Let aₙ=46+8n and bₙ=98+4n be two sequences for natural numbers n≤100. Then, the sum of all terms common to both the sequences is:

Question

UPSC Quizroom · Accepted Answer

Step 1: Find common terms

aₙ = bₘ → 46 + 8n = 98 + 4m

8n − 4m = 52 → 2n − m = 13 → m = 2n − 13

Step 2: Determine valid n and m

n ∈ [1, 100], m ∈ [1, 100]

m ≥ 1 → 2n − 13 ≥ 1 → n ≥ 7

m ≤ 100 → 2n − 13 ≤ 100 → n ≤ 56

So n = 7, 8, …, 56 → total 50 terms

Step 3: Compute sum

Common term = aₙ = 46 + 8n

First term: a₇ = 46 + 56 = 102

Last term: a₅₆ = 46 + 448 = 494

Sum of AP = (number of terms)/2 × (first term + last term)

S = 50/2 × (102 + 494) = 25 × 596 = 14900

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