Question

How many terms of the series27+24+21+18+....will add up to 126?

Answer 1

Either 7 terms or 12 terms of the series will add up to 126.

Step-by-step explanation:

Given, 27 + 24 + 21 + 18 +........

1st term = $a_{1}$ = 27

Common difference = d = 24 - 27 = 21 - 24 = 18 - 21 = -3

Sum = 126, number of terms = n

=> $\frac{n}{2}(2a_{1} + (n - 1)d)$ = 126

$\frac{n}{2}(2\times 27 + (n-1)\times (-3)) = 126\\=> \frac{n}{2}(54 - 3n + 3)=126\\=> \frac{n}{2}(57 - 3n) = 126\\=> 57n - 3n^{2} = 252\\=> 3n^{2} - 57n + 252 = 0\\=> n^{2} - 19n + 84 = 0\\=> n^{2} - 12n - 7n + 84 = 0\\=> n(n - 12) - 7(n - 12) = 0\\=> (n - 12) (n - 7) = 0\\=> n = 12, 7$

Hence, either 7 terms or 12 terms of the series will add up to 126.