If $a_{n} = 3 n - 1$ in an AP where $a_{n}$ is the $n^{t h}$ term, match the following terms with their values.

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Solution

Given that $a_{n} = 3 n - 1.$

The common difference of the AP, $d = a_{2} - a_{1}$
$= 3 (2) - 1 - (3 (1) - 1)$
$= 5 - 2 = 3.$

We know that the $n^{th}$ term of an AP with first term $a$ and common difference $d$ is given by
$a_{n} = a + (n - 1) d .$

$⟹ a_{3} = a + 2 d = 2 + 2 \times 3 = 8,$
$a_{7} = a + 6 d = 2 + 6 \times 3 = 20,$
$a_{13} = a + 12 d = 2 + 12 \times 3 = 38$ and
$a_{8} = a + 7 d = 2 + 7 \times 3 = 23.$
$a_{2} = a + d = 2 + 3 = 5$
$a_{6} = a + 5 d = 2 + 5 \times 3 = 17$

$∴ a_{3} + a_{7} = 8 + 20 = 28$ , $a_{2} + a_{6} = 5 + 17 = 22$ and $a_{13} - a_{8} = 38 - 23 = 15$

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