The $8$ th term of an A.P. is $17$ and the $19$ th term is $39$ . Find the $25$ th term.

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Solution

Given that: $T_{8} = 17, T_{19} = 39, T_{25} = ?$
To find $T_{25}$ , first we should find $a$ and $d$ .
$d = \frac{T_{p} - T_{q}}{p - q}$
$d = \frac{T_{19} - T_{8}}{19 - 8} = \frac{39 - 17}{11} = \frac{22}{11} ∴ d = 2$
Consider,
$T_{8} = 17$
$a + 7 d = 17$ $[∴ T_{n} = a + (n - 1) d]$
$a + 7 (2) = 17$
$a + 14 = 17$
$∴ a = 3$
To find $T_{25}, T_{n} = a + (n - 1) d$
$T_{25} = 3 + (25 - 1) 2 = 3 + (24 \times 2) = 3 + 48$
$∴ T_{25} = 51$

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