Multiple choice

Consider two arithmetic series : $\begin{array} { l } { A _ { 1 } : 2 + 9 + 16 + 23 + \ldots \ldots \ldots + 205 } \ { A _ { 2 } : 5 + 9 + 13 + 17 + \ldots \ldots \ldots + 161 } \end{array}$ then the number of terms common to the two series is

  1. $6$
  2. $8$
  3. $10$
  4. $12$
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A Correct answer
Explanation

A1: 2, 9, 16, ..., 205 (a=2, d=7). A2: 5, 9, 13, ..., 161 (a=5, d=4). Common terms start at 9, with common difference = LCM(7,4) = 28. Series: 9, 37, 65, 93, 121, 149. Next term 177 > 161. Total 6 terms.

AI explanation

The nth term of the first series is given by 7n - 5 and the nth term of the second series is 4m + 1. Set the terms equal to find common values, resulting in 7n - 4m = 6, which implies that the common terms must leave a remainder of 1 when divided by 7 and a remainder of 1 when divided by 4. The common terms therefore form a new arithmetic progression with the first term 9 and a common difference of 28 (the least common multiple of 7 and 4), yielding the sequence 9, 37, 65, 93, 121, 149. To find the number of common terms, use the nth term formula 9 + (k - 1)28 <= 161, which solves to k < 6.64, meaning there are exactly 6 common terms.

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