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
Explanation:
n can take any value from 1 till 99.
∴ an = 54, 62, 70, 78, 86, 94, 102, ..., 838 and bn = 102, 106, 110, ..., 494
These two are arithmetic progressions. Common terms of 2 APs are also in AP whose common difference is LCM of the common difference of original APs. Common difference of an and bn is 8 and 4 respectively. ∴ The common difference of the common terms = LCM(8, 4) = 8
⇒ The common terms are 102, 110, 118, ...
Now, nth term of this series = 102 + 8(n - 1) This should be less than or equal to 494 ⇒ 102 + 8(n - 1) ≤ 494 ⇒ n ≤ 50
Sum of all these 50 terms = 50/2 × (102 + 494) = 14900
Hence, 14900.
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