Question

27. Find the number of terms of the AP 18, 15, 12. ....... so that their sum is 45. Explain the double
answer

Answer 1

${\bold{\underline{\underline{Answer:}}}}$

${\bold{\therefore Number\:of\:terms=3\:and\:10}}$

${\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\underline \bold{Given : } \\ \implies A.P = 18,15,12,... \\ \\ \implies First \: term(a) = 18 \\ \\ \implies Common \: difference(d) = - 3 \\ \\ \implies s_{n} = 45 \\ \\ \underline \bold{ To \: Find : } \\ \implies n = ?$

• According to given question :

$\bold{Using \: formula \: of \: sum \: of \: n_{th} \: term : } \\ \implies s_{n} = \frac{n}{2}(2a + (n - 1) \times d) \\ \\ \implies 45 = \frac{n}{2} (2 \times 18 + (n - 1) \times - 3 \\ \\ \implies 45 \times 2 = n(36 - 3n + 3) \\ \\ \implies 90 = n(39 - 3n) \\ \\ \implies 3 {n}^{2} - 39n + 90 = 0 \\ \\ \implies {n}^{2} - 13n +30 = 0 \\ \\ \implies {n}^{2} - 10n- 3n + 30 = 0 \\ \\ \implies n(n - 10) - 3(n - 10) = 0 \\ \\ \implies n = 3 \: and \: 10 \\ \\ \bold{Note : In \: A.P \: value \: of \: n \: never \: in \: negative} \\ \bold{ \implies n = 3\:and\:10}$

Answer 2

$\large \sf \underline{ \underline{ \sf \: Answer : \: \: \: }}$

$\sf n = 10 \: \: or \: \: n = 3$

$\large \sf \underline{ \underline{ \sf \: \: Explaination : \: \: \: }}$

Given ,

First term (a) = 18

Common difference (d) = -3

Sum of nth term (Sn) = 45

We Know that ,

$\huge{ \star}$ $\sf S_{n} = \frac{n}{2} \bigg (2a + (n - 1)d \bigg)$

$\to \sf 45 = \frac{n}{2} \bigg(2 \times 18 + (n - 1)( - 3) \bigg) \\ \\ \to \sf 90 = n(36 - 3n + 3) \\ \\\to \sf 90 = 39n - 3 {n}^{2} \\ \\\to \sf 3 {n}^{2} - 39n + 90 = 0 \\ \\\to \sf 3( {n}^{2} - 13n + 30) = 0 \\ \\ \to \sf {n}^{2} - 10n - 3n + 30 = 0\\ \\\to \sf n(n - 10) - 3(n - 10) = 0 \\ \\\to \sf (n - 10)(n - 3) = 0 \\ \\\to \sf n = 10 \: \: or \: \: n = 3$

$\therefore$The required value of n is 10 or 3

__________________Tera Bhai ❤