Question

How many terms of the A.P. 27,24,21,… should be taken so that their sum is zero?

Answer 1

AnswEr:-

❀ Your Answer Is 19 Terms.

ExplanaTion:-

Given:-

• An AP.

• Terms of an AP are 27, 24, 21....

To Find:-

• The number of terms which ahould be taken so that their sum would be 0.

Formula Used:-

$\\ \bf\longrightarrow S_n = \dfrac{n}{2}[2a + (n - 1)d].$

Where,

• $\tt S_n$ = Sum of n terms.

• n = Number of terms.

• a = First term of the AP.

• $\tt a_2$ = Second term.

• d = Common Difference $\tt (a_2-a)$.

So Here,

• $\tt S_n$ = 0.

• n = ?? [To Find].

• a = 27.

• $\tt a_2$ = 24.

• $\tt d = 24 - 27 = -3.$

So lets put the value in the above formula:-

$\\ \tt\mapsto \frac{n}{2} [2a + (n - 1)d ] = S_n. \\ \\ \tt\mapsto \frac{n}{2} [2(27) + (n - 1)( - 3)] = 0. \\ \\ \tt\mapsto \frac{n}{2} [54 + 3 - 3n] = 0. \\ \\ \tt\mapsto \frac{n}{2} [57 - 3n] = 0. \\ \\ \tt\mapsto n[3n - 57] = 0 \times 2. \\ \\ \tt\mapsto n = 0 \: \: \: \: or \: \: \: \: 3n - 57 = 0. \\ \\ \bf\mapsto n = 0 \: \: \: \: or \: \: \: \: n = 19.$

So the value of n is 0, which is not possible or 19.

$\small{\boxed{\bf\therefore\:\:The\:\:Number\:\:Of\:\:Terms\:=\:19}}$

Answer 2

Answer:

$Consider \: the \: \: given \: A.P. \: series.</p><p> \\ </p><p>27,24,21,......</p><p></p><p> \\ </p><p>Here, a=27,d=−3</p><p></p><p> \\ </p><p>Since, Sum=0</p><p> \\ </p><p></p><p>Therefore,</p><p></p><p>$

$sum = \frac{n}{2} [2a + (n - 1)d]$

$0= \frac{n}{2} [2 \times 27 + (n - 1) \times - 3]$

$54 - 3n + 3 = 0$

$57 - 3n = 0$

$57 = 3n$

$n = \frac{ \cancel{ 57}}{ \cancel 3} = 19$

$so, \: \boxed{n = 19}$