Let an and bn be two sequences such that an = 13 + 6(n - 1) and bn = 15 + 7(n - 1) for all natural numbers n. Then, the largest three digit integer that is common to both these sequences is
Explanation:
The given series are APs
Series an is: 13, 19, 25, 31, 37, 43, 49, ... Series bn is: 15, 22, 29, 36, 43, 50, ...
For 2 APs, their common terms are also in AP, with common difference as LCM of common difference of the orignal 2 APs. The first common term of the two series is 43 and the common difference of the two series is LCM (6, 7) = 42 ∴ The series comprising of common terms is 43, 85, 127, ...
Now, nth term of this series = 43 + 42(n - 1) = 42n + 1
⇒ 42n + 1 < 1000 ⇒ n < 999/42 = 23.78 ∴ Highest possible value of n = 23
⇒ Highest three-digit term common to both the original series = 42 × 23 + 1 = 967.
Hence, 967.
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