Question

the sum n terms of an ap sequence is 203. the first is 20 and the common difference is 3 . find the number of terms n in the ap sequence ​

Answer 1

Solution:-

Given:-

$\to \rm\: S_n = 203$

$\to \: \rm \:first \: term \: ( a) = \: 20$

$\to \rm \:common \: \: difference \: ( d) \: = 3$

$\boxed{ \green { \bf \: S_n = \frac{n}{2} \{2a \: + (n - 1)d \} }}$

Put the value on formula

$\rm \to \: 203 = \frac{n}{2} \{2 \times 20 + (n - 1)3 \}$

$\rm \: \to \: 203 = \frac{n}{2} \{40 + 3n - 3 \}$

$\rm \: \to \: 203 = \frac{n}{2} \{37 + 3n \}$

$\rm \to \: 406 = 37n + 3 {n}^{2}$

$\rm \to \: 3 {n}^{2} + 37n - 406 = 0$

$\to \rm \: 3 {n}^{2} + 58n - 21n - 406 = 0$

$\rm \to \: n(3n + 58) - 7(3n + 58) = 0$

$\rm \to \: (n - 7)(3n + 58) = 0$

$\rm \to \: n \: = 7 \: \: and \: \: \frac{58}{3}$

$\to \rm \: n = 7$

n = 58/3 is not nth term because n is not in the point form