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Formula for Sum of n Terms of an AP

In an AP, giv...

Question

In an AP, given $a_{n} = 4, d = 2, S_{n} = - 14,$ find $n$ and $a$ .

A

n = 11

and

a = - 3

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B

n = 10

and

a = - 3

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C

n = 7

and

a = - 8

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D

n = 10

and

a = 3

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Solution

The correct option is C $n = 7$ and $a = - 8$
As we know nth term, $a_{n} = a + (n - 1) d$
& Sum of first $n$ terms, $S_{n} = \frac{n}{2} (2 a + (n - 1) d)$ , where $a$ & $d$ are the first term amd common difference of an AP.
Since, $a_{n} = 4 \Rightarrow a + (n - 1) d = 4$
$\Rightarrow a + (n - 1) 2 = 4$
$\Rightarrow a = 6 - 2 n$ ...(1)
Also, $S_{n} = - 14$
$\Rightarrow \frac{n}{2} [2 a + (n - 1) d] = - 14$
$\Rightarrow \frac{n}{2} [2 (6 - 2 n) + (n - 1) 2] = - 14$ ( $∵ f r o m (1)$ )
$\Rightarrow n [12 - 4 n + 2 n - 2] = - 28$
$\Rightarrow n [n - 5] = 14$
$\Rightarrow n^{2} - 5 n - 14 = 0$
$\Rightarrow (n - 7) (n + 2) = 0$
$\Rightarrow n = 7, - 2$
But, $n = - 2$ is not possible.
$∴ n = 7$
Putting $n = 7$ in (1), we get
$∴ a = 6 - 2 \times 7 = - 8$

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Similar questions

Q. In an AP:
(i) Given

a = 5, d = 3

a_{n} = 50

n

S_{n}

a = 7

a_{13} = 35

d

S_{13}

a_{12} = 37

d = 3

a

S_{12}

a_{3} = 15

S_{10} = 125

d

a_{10}

d = 5

S_{9} = 75

a

a_{9}

a = 2, d = 8

S_{n} = 90

n

a_{n}

a = 8

a_{n} = 62

S_{n} = 210

n

d

a_{n} = 4

d = 2

S_{n} = - 14

n

a

a = 3, n = 8, S = 192

d

l = 28, S = 144

, and there are total 9 terms. Find

a

Q. Let there be an A.P. with first term 'a', common difference 'd'. If a_n denotes in n^th term and S_n the sum of first n terms, find.

(i) n and S_n, if a = 5, d = 3 and a_n = 50.

(ii) n and a, if a_n = 4, d = 2 and S_n= −14.

(iii) d, if a = 3, n = 8 and S_n = 192.

(iv) a, if a_n = 28, S_n = 144 and n= 9.

(v) n and d, if a = 8, a_n = 62 and S_n = 210

(vi) n and a_n, if a= 2, d = 8 and S_n = 90.

In an AP

(i) Given $a = 5, d = 3, a_{n} = 50$ , find $n$ and $S_{n}$ .

(ii) Given $a = 7, a_{13} = 35$ , find $d$ and $S_{13}$ .

(iii) Given $a_{12} = 37, d = 3$ , find $a$ and $S_{12}$ .

(iv) Given $a_{3} = 15, S_{10} = 125$ , find $d$ and $a_{10}$ .

(v) Given $d = 5, S_{9} = 75$ , find $a$ and $a_{9}$ .

(vi) Given $a = 2, d = 8, S_{n} = 90$ , find $n$ and $a_{n}$ .

(vii) Given $a = 8, a_{n} = 62, S_{n} = 210$ , find $n$ and $d$ .

(viii) Given $a_{n} = 4, d = 2, S_{n} = - 14$ , find $n$ and $a$ .

(ix) Given $a = 3, n = 8, S_{8} = 192$ , find $d$ .

(x) Given $l = 28, S_{n} = 144$ and there are total $9$ terms. Find $a$ .

Q. Question 3 (viii)
In an AP:
(viii) Given

a_{n} = 4

S_{n} = - 14

, find n and a.

Q. Let there be an A.P. with first term 'a', common difference 'd'. If a_n denotes in n^th term and S_n the sum of first n terms, find.

(i) n and S_n, if a = 5, d = 3 and a_n = 50.

(ii) n and a, if a_n = 4, d = 2 and S_n= −14.

(iii) d, if a = 3, n = 8 and S_n = 192.

(iv) a, if a_n = 28, S_n = 144 and n= 9.

(v) n and d, if a = 8, a_n = 62 and S_n = 210

(vi) n and a_n, if a= 2, d = 8 and S_n = 90.

(vii)

k, if S_{n} = 3 n^{2} + 5 n and a_{k} = 164

S_{22}, if d = 22 and a_{22} = 149

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