Algebra

Progression & Series

Medium

Let $S$ denote an arithmetic progression whose first term is either 132 or 158, and the common difference is an even integer less than 10. If the $n^{\text{th}}$ term of $S$ is 174, then the number of possible distinct values of $n$ is ___

Entered answer:

✅ Correct Answer: 5

The $n$ -th term of an AP: $a + (n - 1)d = 174$ , so $(n - 1)d = 174 - a$ .

The common difference $d$ must be a positive even integer less than $10$ , so $d \in \{2, 4, 6, 8\}$ . ( $d$ must be positive because $174$ is greater than both possible first terms.)

Case $a = 132$ : $(n - 1)d = 42$ .

$d = 2 \Rightarrow n = 22$ .

$d = 4 \Rightarrow n - 1 = 10.5$ , not an integer.

$d = 6 \Rightarrow n = 8$ .

$d = 8 \Rightarrow n - 1 = 5.25$ , not an integer.

Case $a = 158$ : $(n - 1)d = 16$ .

$d = 2 \Rightarrow n = 9$ .

$d = 4 \Rightarrow n = 5$ .

$d = 6 \Rightarrow n - 1 = \dfrac{8}{3}$ , not an integer.

$d = 8 \Rightarrow n = 3$ .

All distinct values of $n$ : $\{3, 5, 8, 9, 22\}$ .

Number of distinct values $= 5$ .

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Let $S$ denote an arithmetic progression whose first term is either 132 or 158, and the common difference is an even integer less than 10. If the $n^{\text{th}}$ term of $S$ is 174, then the number of possible distinct values of $n$ is ___