Multiple choice

The nth terms of two sequences are defined as follows: an = 11 + 4n and An = 9 + 5n. If the last term of both sequences is less than 500, find the number of terms that these sequences have in common.

  1. 12

  2. 18

  3. 25

  4. 28

  5. None of these
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C Correct answer
Explanation

a_n = 11 + 4n, A_m = 9 + 5m. 11 + 4n = 9 + 5m -> 5m = 4n + 2. For m to be an integer, 4n+2 must be divisible by 5. n=2 (m=2, value=19), n=7 (m=6, value=39), n=12 (m=10, value=59). n follows arithmetic progression with d=5. Last term < 500: 11 + 4n < 500 -> 4n < 489 -> n <= 122. Terms: 2, 7, 12, ..., 122. Number of terms = (122 - 2)/5 + 1 = 25.

AI explanation

We are given two arithmetic progressions with general terms an = 11 + 4n and An = 9 + 5n. Equating the two general terms gives 11 + 4n = 9 + 5m, which simplifies to 5m - 4n = 2. Testing integer values for m from 1 upwards reveals that common terms occur at n = 2, 7, 12, and so forth, forming a new progression with a common difference of 5. Since the last term must be less than 500, the maximum value for n in the sequence an = 11 + 4n is 122, meaning the valid values of n that produce common terms are 2, 7, 12, up to 122, resulting in exactly 25 common terms. The correct result is 25.

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