**Find the Indicated Terms of the Arithmetic Sequence :**

In this section, let us see some example problems of finding the missing term from the given information.

To find the sum of an arithmetic series, we use the formulas give below.

S_{n} = (n/2) [a + l] (or)

S_{n} = (n/2) [2a + (n - 1)d]

n^{th} term of arithmetic progression :

a_{n} = a + (n - 1)d

Total number of terms :

n = [(l - a)/d] + 1

a = first term, d = common difference, n = number of terms and l = last term.

**Question 1 :**

Given a_{n} = 4, d = 2, S_{n} = -14 find n and a

**Solution :**

a_{n} = 4

a + (n - 1) d = 4

a + (n - 1)2 = 4

a + 2n - 2 = 4

a + 2n = 6

a = 6 - 2n ---(1)

S_{n} = -14

S_{n} = (n/2) [2a + (n - 1)d]

-14 = (n/2) [2a + (n - 1)2]

-14 = (n/2) [2a + 2n - 2] ----(2)

By applying the value of a in (2), we get

-14 = (n/2)[2(6 - 2n) + 2n - 2]

-28 = n[12 - 4n + 2n - 2]

-28 = n[10 - 2n]

-28 = 10n - 2n^{2}

2n^{2} - 10n - 28 = 0

Dividing it by 2, we get

n^{2} - 5n - 14 = 0

(n - 7) (n + 2) = 0

n = 7 or n = -2 (not admissible)

By applying the value of n in (1), we get

a = 6 - 2(7)

a = 6 - 14

a = -8

Hence the values of a and n are -8 and 7 respectively.

**Question 2 :**

Given a = 3, n = 8, S_{n} = 192 find d

**Solution :**

S_{n }= 192

S_{n} = (n/2) [2a + (n - 1)d]

192 = (8/2) [2(3) + (8 - 1)d]

192 = 4 [6 + 7d]

192/4 = 6 + 7d

48 = 6 + 7d

48 - 6 = 7d

7d = 42

d = 42/7 = 6

Hence the value of d is 6.

**Question 3 :**

Given l = 28 , S = 144 , and there are total 9 terms. Find a

**Solution :**

Let "l" be the last term or n^{th} term of the series.

l = t_{n} = 28

t_{n} = a + (n - 1)d

a + (9 - 1)d = 28

a + 8d = 28 ----(1)

S_{9} = 144

S_{n} = (n/2) [2a + (n - 1)d]

(n/2) [2a + (n - 1)d] = 144

(9/2) [2a + 8d] = 144

[2a + 8d] = 144 ⋅ (2/9)

2a + 8d = 32

a + 4d = 16 ------(2)

(1) - (2)

a + 8d = 28

a + 4d = 16

(-) (-) (-)

---------------

4d = 12

d = 3

By applying the value of d in (1), we get

a + 8(3) = 28

a + 24 = 28

a = 28 - 24 = 4

Hence the first term is 4.

After having gone through the stuff given above, we hope that the students would have understood, find the indicated terms of the arithmetic sequence.

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