{
  "title": "Sum_of_first_n_terms(Arithmetic_Progressions) - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Formula for sum of first n terms of AP is:", "options": {"A": "S_n = n/2[2a+(n-1)d]", "B": "S_n = n/2[a+l]", "C": "Both A and B", "D": "S_n = a+(n-1)d"}, "correct_answer": "C"},
    {"id": 2, "difficulty": "easy", "question": "For AP: 2,4,6,8,..., find sum of first 5 terms.", "options": {"A": "20", "B": "25", "C": "30", "D": "35"}, "correct_answer": "C"},
    {"id": 3, "difficulty": "easy", "question": "If a=3, d=2, n=10, find S_n.", "options": {"A": "100", "B": "110", "C": "120", "D": "130"}, "correct_answer": "C"},
    {"id": 4, "difficulty": "easy", "question": "Sum of first 20 natural numbers is:", "options": {"A": "190", "B": "200", "C": "210", "D": "220"}, "correct_answer": "C"},
    {"id": 5, "difficulty": "easy", "question": "For AP: 5,10,15,..., find sum of first 8 terms.", "options": {"A": "160", "B": "170", "C": "180", "D": "190"}, "correct_answer": "C"},
    {"id": 6, "difficulty": "easy", "question": "If a=1, l=10, n=10, find S_n using S_n=n/2(a+l).", "options": {"A": "50", "B": "55", "C": "60", "D": "65"}, "correct_answer": "B"},
    {"id": 7, "difficulty": "easy", "question": "Sum of first 100 positive integers:", "options": {"A": "4950", "B": "5000", "C": "5050", "D": "5100"}, "correct_answer": "C"},
    {"id": 8, "difficulty": "easy", "question": "For AP: 7,14,21,..., find sum of first 6 terms.", "options": {"A": "137", "B": "147", "C": "157", "D": "167"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "If a=10, d=-1, n=5, find S_n.", "options": {"A": "40", "B": "45", "C": "50", "D": "55"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "Sum of first 15 even numbers:", "options": {"A": "210", "B": "220", "C": "230", "D": "240"}, "correct_answer": "D"},
    {"id": 11, "difficulty": "easy", "question": "For AP: 3,7,11,..., find sum of first 4 terms.", "options": {"A": "36", "B": "40", "C": "44", "D": "48"}, "correct_answer": "B"},
    {"id": 12, "difficulty": "easy", "question": "If S_n=45, n=10, a=0, find d.", "options": {"A": "0.5", "B": "1", "C": "1.5", "D": "2"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "Sum of first 12 multiples of 5:", "options": {"A": "360", "B": "370", "C": "380", "D": "390"}, "correct_answer": "D"},
    {"id": 14, "difficulty": "easy", "question": "For AP: 1,3,5,..., find sum of first 10 terms.", "options": {"A": "90", "B": "95", "C": "100", "D": "105"}, "correct_answer": "C"},
    {"id": 15, "difficulty": "easy", "question": "If a=2, d=3, find S_5.", "options": {"A": "35", "B": "40", "C": "45", "D": "50"}, "correct_answer": "B"},
    {"id": 16, "difficulty": "easy", "question": "Sum of first 8 odd numbers:", "options": {"A": "60", "B": "62", "C": "64", "D": "66"}, "correct_answer": "C"},
    {"id": 17, "difficulty": "easy", "question": "For AP: 20,18,16,..., find sum of first 5 terms.", "options": {"A": "80", "B": "85", "C": "90", "D": "95"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "If a=100, d=-5, n=5, find S_n.", "options": {"A": "450", "B": "460", "C": "470", "D": "480"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "Sum of first 25 natural numbers:", "options": {"A": "300", "B": "315", "C": "325", "D": "350"}, "correct_answer": "C"},
    {"id": 20, "difficulty": "easy", "question": "For AP: ½, 1, 1½,..., find sum of first 6 terms.", "options": {"A": "9", "B": "10", "C": "10.5", "D": "11"}, "correct_answer": "C"},
    {"id": 21, "difficulty": "medium", "question": "Find sum of first 24 terms of AP: 5,8,11,14,...", "options": {"A": "900", "B": "924", "C": "948", "D": "972"}, "correct_answer": "B"},
    {"id": 22, "difficulty": "medium", "question": "If sum of first n terms is S_n=2n²+3n, find a and d.", "options": {"A": "a=5,d=4", "B": "a=3,d=4", "C": "a=5,d=2", "D": "a=3,d=2"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "How many terms of AP: 9,17,25,... must be taken to give sum 636?", "options": {"A": "10", "B": "11", "C": "12", "D": "13"}, "correct_answer": "C"},
    {"id": 24, "difficulty": "medium", "question": "Find sum of first 40 positive integers divisible by 6.", "options": {"A": "4800", "B": "4920", "C": "5040", "D": "5160"}, "correct_answer": "B"},
    {"id": 25, "difficulty": "medium", "question": "If S_7=49 and S_17=289, find S_n formula.", "options": {"A": "S_n=n²", "B": "S_n=2n²", "C": "S_n=3n²", "D": "S_n=4n²"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "Find sum of integers between 100 and 200 divisible by 9.", "options": {"A": "1584", "B": "1683", "C": "1782", "D": "1881"}, "correct_answer": "B"},
    {"id": 27, "difficulty": "medium", "question": "If sum of first 15 terms is 375 and first term is 5, find common difference.", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "B"},
    {"id": 28, "difficulty": "medium", "question": "Find sum of first 51 terms of AP whose 2nd term is 2 and 4th term is 8.", "options": {"A": "2550", "B": "2601", "C": "2652", "D": "2703"}, "correct_answer": "B"},
    {"id": 29, "difficulty": "medium", "question": "How many terms of AP: 63,60,57,... must be taken so sum is 693?", "options": {"A": "20", "B": "21", "C": "22", "D": "23"}, "correct_answer": "C"},
    {"id": 30, "difficulty": "medium", "question": "If S_n=5n²+3n, find a_10.", "options": {"A": "103", "B": "105", "C": "107", "D": "109"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "Find sum of all 3-digit natural numbers divisible by 7.", "options": {"A": "70336", "B": "70371", "C": "70406", "D": "70441"}, "correct_answer": "B"},
    {"id": 32, "difficulty": "medium", "question": "If sum of first 7 terms is 49 and sum of first 17 terms is 289, find sum of first n terms.", "options": {"A": "n²", "B": "2n²", "C": "3n²", "D": "n(n+1)"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "Find sum of first 30 terms of AP: 10,7,4,...", "options": {"A": "-615", "B": "-625", "C": "-635", "D": "-645"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "How many terms of AP: 45,39,33,... give sum 180?", "options": {"A": "8", "B": "9", "C": "10", "D": "11"}, "correct_answer": "C"},
    {"id": 35, "difficulty": "medium", "question": "If S_5=35 and S_10=120, find a and d.", "options": {"A": "a=3,d=2", "B": "a=4,d=2", "C": "a=3,d=3", "D": "a=4,d=3"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "Find sum of all natural numbers between 250 and 1000 divisible by 3.", "options": {"A": "156375", "B": "157375", "C": "158375", "D": "159375"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "If sum of first n terms is 3n²+5n, find a_n.", "options": {"A": "6n+2", "B": "6n+3", "C": "6n+4", "D": "6n+5"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "Find sum of first 25 terms of AP whose nth term is 7-3n.", "options": {"A": "-725", "B": "-750", "C": "-775", "D": "-800"}, "correct_answer": "D"},
    {"id": 39, "difficulty": "medium", "question": "How many terms of AP: -6,-11/2,-5,... give sum -25?", "options": {"A": "10", "B": "12", "C": "14", "D": "16"}, "correct_answer": "B"},
    {"id": 40, "difficulty": "medium", "question": "If S_m=S_n for m≠n, prove S_{m+n}=0. For a=2,d=3, find m,n with S_m=S_n.", "options": {"A": "m=4,n=6", "B": "m=5,n=7", "C": "m=6,n=8", "D": "m=7,n=9"}, "correct_answer": "B"},
    {"id": 41, "difficulty": "medium", "question": "Find sum of first 20 terms of AP: 0.5,1.0,1.5,...", "options": {"A": "100", "B": "105", "C": "110", "D": "115"}, "correct_answer": "B"},
    {"id": 42, "difficulty": "medium", "question": "If S_12=144 and a=3, find d.", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 43, "difficulty": "medium", "question": "Find sum of all 2-digit numbers divisible by 4.", "options": {"A": "1152", "B": "1188", "C": "1224", "D": "1260"}, "correct_answer": "B"},
    {"id": 44, "difficulty": "medium", "question": "If S_n=n(4n+1), find a_15.", "options": {"A": "117", "B": "119", "C": "121", "D": "123"}, "correct_answer": "B"},
    {"id": 45, "difficulty": "medium", "question": "Find sum of first 15 multiples of 8.", "options": {"A": "840", "B": "880", "C": "920", "D": "960"}, "correct_answer": "D"},
    {"id": 46, "difficulty": "medium", "question": "If sum of first p terms = sum of first q terms (p≠q), find sum of first (p+q) terms.", "options": {"A": "0", "B": "a(p+q)", "C": "p+q", "D": "pq"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "medium", "question": "Find sum of first 40 terms of AP: 3,6,9,...", "options": {"A": "2400", "B": "2460", "C": "2520", "D": "2580"}, "correct_answer": "B"},
    {"id": 48, "difficulty": "medium", "question": "If S_8=100 and S_16=400, find a and d.", "options": {"A": "a=2,d=3", "B": "a=3,d=2", "C": "a=4,d=3", "D": "a=3,d=4"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "medium", "question": "Find sum of all odd numbers between 0 and 100.", "options": {"A": "2400", "B": "2450", "C": "2500", "D": "2550"}, "correct_answer": "C"},
    {"id": 50, "difficulty": "medium", "question": "If S_n=an²+bn, prove it's AP. Find a and d in terms of a,b.", "options": {"A": "a=a+b, d=2a", "B": "a=b, d=2a", "C": "a=a, d=2a", "D": "a=b-a, d=2a"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "medium", "question": "Find sum of first 50 terms of AP: 1,3,5,...", "options": {"A": "2400", "B": "2450", "C": "2500", "D": "2550"}, "correct_answer": "C"},
    {"id": 52, "difficulty": "medium", "question": "If sum of first n terms is n(n+1), find 10th term.", "options": {"A": "18", "B": "20", "C": "22", "D": "24"}, "correct_answer": "B"},
    {"id": 53, "difficulty": "medium", "question": "Find sum of integers from 1 to 100 divisible by 2 or 5.", "options": {"A": "3050", "B": "3100", "C": "3150", "D": "3200"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "medium", "question": "If S_10=120 and S_20=440, find S_30.", "options": {"A": "960", "B": "980", "C": "1000", "D": "1020"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "medium", "question": "Find sum of first 12 terms of AP: √2, 2√2, 3√2,...", "options": {"A": "72√2", "B": "75√2", "C": "78√2", "D": "81√2"}, "correct_answer": "C"},
    {"id": 56, "difficulty": "medium", "question": "If S_n=3n²+2n, find d.", "options": {"A": "4", "B": "5", "C": "6", "D": "7"}, "correct_answer": "C"},
    {"id": 57, "difficulty": "medium", "question": "Find sum of first 100 natural numbers divisible by 5.", "options": {"A": "25250", "B": "25300", "C": "25350", "D": "25400"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "medium", "question": "If S_p=S_q (p≠q), prove a+(p+q-1)d=0. For a=5,d=-1, find p,q.", "options": {"A": "p=6,q=10", "B": "p=7,q=11", "C": "p=8,q=12", "D": "p=9,q=13"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "medium", "question": "Find sum of first 25 terms of AP: 4,7,10,...", "options": {"A": "950", "B": "975", "C": "1000", "D": "1025"}, "correct_answer": "C"},
    {"id": 60, "difficulty": "medium", "question": "If S_4=16 and S_8=64, find S_12.", "options": {"A": "144", "B": "156", "C": "168", "D": "180"}, "correct_answer": "A"},
    {"id": 61, "difficulty": "medium", "question": "Find sum of all 3-digit numbers divisible by 11.", "options": {"A": "44550", "B": "44605", "C": "44660", "D": "44715"}, "correct_answer": "A"},
    {"id": 62, "difficulty": "medium", "question": "If S_n=2n²+5n, find a_20.", "options": {"A": "83", "B": "85", "C": "87", "D": "89"}, "correct_answer": "A"},
    {"id": 63, "difficulty": "medium", "question": "Find sum of first 30 odd numbers.", "options": {"A": "870", "B": "900", "C": "930", "D": "960"}, "correct_answer": "B"},
    {"id": 64, "difficulty": "medium", "question": "If ratio S_m:S_n=m²:n², find ratio a_m:a_n.", "options": {"A": "(2m-1):(2n-1)", "B": "(2m+1):(2n+1)", "C": "m:n", "D": "m²:n²"}, "correct_answer": "A"},
    {"id": 65, "difficulty": "medium", "question": "Find sum of first 40 even numbers.", "options": {"A": "1620", "B": "1640", "C": "1660", "D": "1680"}, "correct_answer": "B"},
    {"id": 66, "difficulty": "medium", "question": "If S_6=42 and a=2, find l (last term of first 6 terms).", "options": {"A": "12", "B": "13", "C": "14", "D": "15"}, "correct_answer": "A"},
    {"id": 67, "difficulty": "medium", "question": "Find sum of numbers between 200 and 400 divisible by 7.", "options": {"A": "8729", "B": "8772", "C": "8815", "D": "8858"}, "correct_answer": "A"},
    {"id": 68, "difficulty": "medium", "question": "If S_n=4n²+3n, find a_15.", "options": {"A": "119", "B": "123", "C": "127", "D": "131"}, "correct_answer": "B"},
    {"id": 69, "difficulty": "medium", "question": "Find sum of first 20 terms of AP: 1/2, 3/2, 5/2,...", "options": {"A": "190", "B": "200", "C": "210", "D": "220"}, "correct_answer": "B"},
    {"id": 70, "difficulty": "medium", "question": "If S_3=15 and S_6=45, find S_9.", "options": {"A": "90", "B": "95", "C": "100", "D": "105"}, "correct_answer": "A"},
    {"id": 71, "difficulty": "medium", "question": "Find sum of all natural numbers ≤100 divisible by 6.", "options": {"A": "816", "B": "846", "C": "876", "D": "906"}, "correct_answer": "A"},
    {"id": 72, "difficulty": "medium", "question": "If S_n=n(3n+1)/2, find a_10.", "options": {"A": "29", "B": "30", "C": "31", "D": "32"}, "correct_answer": "C"},
    {"id": 73, "difficulty": "medium", "question": "Find sum of first 50 terms of AP: 2,5,8,...", "options": {"A": "3725", "B": "3775", "C": "3825", "D": "3875"}, "correct_answer": "B"},
    {"id": 74, "difficulty": "medium", "question": "If S_5=75 and S_10=300, find a and d.", "options": {"A": "a=3,d=6", "B": "a=5,d=6", "C": "a=3,d=5", "D": "a=5,d=5"}, "correct_answer": "A"},
    {"id": 75, "difficulty": "medium", "question": "Find sum of first 100 terms of AP: 1,2,3,...", "options": {"A": "4950", "B": "5000", "C": "5050", "D": "5100"}, "correct_answer": "C"},
    {"id": 76, "difficulty": "medium", "question": "If S_m=n and S_n=m (m≠n), find S_{m+n}.", "options": {"A": "-(m+n)", "B": "0", "C": "m+n", "D": "mn"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "medium", "question": "Find sum of first 35 terms of AP: 10,6,2,...", "options": {"A": "-1470", "B": "-1490", "C": "-1510", "D": "-1530"}, "correct_answer": "A"},
    {"id": 78, "difficulty": "medium", "question": "If S_n=5n²+2n, find common difference.", "options": {"A": "8", "B": "9", "C": "10", "D": "11"}, "correct_answer": "C"},
    {"id": 79, "difficulty": "medium", "question": "Find sum of all 2-digit odd numbers.", "options": {"A": "2425", "B": "2450", "C": "2475", "D": "2500"}, "correct_answer": "C"},
    {"id": 80, "difficulty": "medium", "question": "If S_4=26 and S_8=100, find S_12.", "options": {"A": "210", "B": "222", "C": "234", "D": "246"}, "correct_answer": "C"},
    {"id": 81, "difficulty": "hard", "question": "If ratio of sums of m and n terms is m²:n², prove that ratio of mth and nth terms is (2m-1):(2n-1). For a=2,d=3, verify for m=3,n=4.", "options": {"A": "7:11", "B": "8:11", "C": "7:10", "D": "8:10"}, "correct_answer": "B"},
    {"id": 82, "difficulty": "hard", "question": "If S_m=S_n for m≠n, prove that S_{m+n}=0. Find condition on a,d for this.", "options": {"A": "a+(m+n-1)d/2=0", "B": "a+(m+n-1)d=0", "C": "2a+(m+n-1)d=0", "D": "a+(m+n)d=0"}, "correct_answer": "C"},
    {"id": 83, "difficulty": "hard", "question": "If sum of first p,q,r terms of AP are a,b,c respectively, prove that a/p(q-r)+b/q(r-p)+c/r(p-q)=0.", "options": {"A": "Always true", "B": "True for specific values", "C": "Never true", "D": "True only if p=q=r"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "Find sum of first n terms of AP whose kth term is 5k+1. Then find sum of first 20 terms.", "options": {"A": "S_n=(5n²+7n)/2, S_20=1070", "B": "S_n=(5n²+7n)/2, S_20=1140", "C": "S_n=(5n²+9n)/2, S_20=1070", "D": "S_n=(5n²+9n)/2, S_20=1140"}, "correct_answer": "A"},
    {"id": 85, "difficulty": "hard", "question": "If S_n,S_{2n},S_{3n} are sums of first n,2n,3n terms of AP, prove that S_{3n}=3(S_{2n}-S_n). Verify for AP 1,3,5,... with n=2.", "options": {"A": "S_6=36,3(S_4-S_2)=36", "B": "S_6=36,3(S_4-S_2)=27", "C": "S_6=27,3(S_4-S_2)=27", "D": "S_6=27,3(S_4-S_2)=36"}, "correct_answer": "A"},
    {"id": 86, "difficulty": "hard", "question": "If sum of n terms is S_n=pn+qn², find a and d in terms of p,q.", "options": {"A": "a=p+q, d=2q", "B": "a=p, d=2q", "C": "a=p+q, d=q", "D": "a=p, d=q"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "Prove that sum of n terms of AP with first term a and last term l is S_n=n(a+l)/2. Then derive S_n=n/2[2a+(n-1)d].", "options": {"A": "Both derivations valid", "B": "Only first valid", "C": "Only second valid", "D": "Neither valid"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "If S_1,S_2,S_3 are sums of first n,2n,3n terms of AP, prove that S_1,S_2-S_1,S_3-S_2 are in AP. For AP 2,5,8,... with n=3, verify.", "options": {"A": "18,45,72 in AP", "B": "18,45,73 not in AP", "C": "18,46,72 not in AP", "D": "18,46,73 in AP"}, "correct_answer": "A"},
    {"id": 89, "difficulty": "hard", "question": "Find sum of series: 1²+(1²+2²)+(1²+2²+3²)+... to n terms. Hint: Use AP concepts.", "options": {"A": "n(n+1)²(n+2)/12", "B": "n(n+1)(n+2)(n+3)/12", "C": "n(n+1)²(n+2)/6", "D": "n(n+1)(n+2)(n+3)/6"}, "correct_answer": "A"},
    {"id": 90, "difficulty": "hard", "question": "If sum of first n terms is S_n and sum of first n terms of another AP is T_n, and ratio S_n:T_n=(7n+1):(4n+27), find ratio of their 11th terms.", "options": {"A": "3:4", "B": "4:5", "C": "5:6", "D": "6:7"}, "correct_answer": "B"},
    {"id": 91, "difficulty": "hard", "question": "Prove that sum of n arithmetic means between a and b is n(a+b)/2. For a=2,b=8 with 3 AMs, verify.", "options": {"A": "Sum=15, n(a+b)/2=15", "B": "Sum=15, n(a+b)/2=20", "C": "Sum=20, n(a+b)/2=15", "D": "Sum=20, n(a+b)/2=20"}, "correct_answer": "A"},
    {"id": 92, "difficulty": "hard", "question": "If S_p=S_q (p≠q), prove that sum of first (p+q) terms is 0. For AP with a=3,d=-1, find p,q such that S_p=S_q.", "options": {"A": "p=4,q=6", "B": "p=5,q=7", "C": "p=6,q=8", "D": "p=7,q=9"}, "correct_answer": "B"},
    {"id": 93, "difficulty": "hard", "question": "Find sum of all natural numbers between 100 and 1000 which leave remainder 2 when divided by 5.", "options": {"A": "98450", "B": "98750", "C": "99050", "D": "99350"}, "correct_answer": "A"},
    {"id": 94, "difficulty": "hard", "question": "If S_n denotes sum of first n terms of AP, prove that S_{12}=3(S_8-S_4). For AP 1,4,7,... verify.", "options": {"A": "S_12=174,3(S_8-S_4)=174", "B": "S_12=174,3(S_8-S_4)=162", "C": "S_12=162,3(S_8-S_4)=174", "D": "S_12=162,3(S_8-S_4)=162"}, "correct_answer": "A"},
    {"id": 95, "difficulty": "hard", "question": "If sum of n terms is S_n=An²+Bn, find ratio of sums of first n terms and next n terms.", "options": {"A": "(2n-1):(2n+1)", "B": "(2n+1):(2n-1)", "C": "n:(n+1)", "D": "(n+1):n"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "hard", "question": "Prove that sum of first n odd numbers is n². Then find sum of odd numbers from 1 to 199.", "options": {"A": "10000", "B": "10201", "C": "10404", "D": "10609"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "hard", "question": "If S_m=n and S_n=m (m≠n), prove that S_{m+n}=-(m+n). For AP with a=5,d=-2, find m,n with S_5=S_7.", "options": {"A": "m=5,n=7 gives S_12=-12", "B": "m=5,n=7 gives S_12=-14", "C": "m=5,n=7 gives S_12=-16", "D": "m=5,n=7 gives S_12=-18"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "hard", "question": "Find sum of series: 5+55+555+... to n terms. Hint: Write as 5/9(9+99+999+...).", "options": {"A": "5/81(10^{n+1}-9n-10)", "B": "5/81(10^{n+1}-10n-9)", "C": "5/81(10^n-9n-10)", "D": "5/81(10^n-10n-9)"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "hard", "question": "If sum of p terms is q and sum of q terms is p (p≠q), prove that sum of (p+q) terms is -(p+q). Find such AP with p=5,q=8.", "options": {"A": "AP: 13,12,11,...", "B": "AP: 13,11,9,...", "C": "AP: 14,13,12,...", "D": "AP: 14,12,10,..."}, "correct_answer": "A"},
    {"id": 100, "difficulty": "hard", "question": "Prove that sum of first n terms of AP is S_n=n/2[2a+(n-1)d]. Then derive alternative form S_n=n/2(a+l).", "options": {"A": "Both forms equivalent", "B": "Only first form correct", "C": "Only second form correct", "D": "Neither form complete"}, "correct_answer": "A"}
  ]
}