**How to Find the Missing Term of an 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 d = 5 , S_{9} = 75 find a and a_{9}

**Solution :**

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

By applying the values of n and d, we get

S_{9} = (9/2) [2a + (9 - 1)5]

75 = (9/2) [2a + 40]

2a + 40 = 150/9

2a = -210/9

a = -35/3

According to the question, we find the 9th term.

9^{th} term :

a_{9} = a + 8d

a_{9} = (-35/3) + 8(5)

a_{9} = (-35/3) + 40

a_{9} = 85/3

Hence, the values of a and a_{9} are -35/3 and 85/3 respectively.

**Question 2 :**

Given a = 2, d = 8, S_{n} = 90 find n and a_{n}

**Solution :**

S_{n} = 90

(n/2) [2a + (n - 1)d] = 90

Applying the values of a and d, we get

(n/2) [2(2) + (n - 1)(8)] = 90

(n/2) [4 + 8n - 8] = 90

(n/2) [8n - 4] = 90

n [4n - 2] = 90

Dividing both sides by 2, we get

n[2n - 1] = 45

2n^{2} - n - 45 = 0

2n^{2} - 10n + 9n - 45 = 0

2n(n - 5) + 9(n - 5) = 0

(n - 5) (2n + 9) = 0

By solving for n, we will get n = 5 and n = -9/2.

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

= 2 + (n - 1)(8)

= 2 + 8n - 8

a_{n }= 8n - 6

Hence, the values of n and a_{n} are 5 and 8n - 6 respectively.

**Question 3 :**

Given a = 8, a_{n} = 62, S_{n} = 210 find n and d

**Solution :**

n^{th} term :

a_{n} = 62

a + (n - 1) d = 62

By applying the value of a, we get

8 + (n - 1) d = 62

(n - 1)d = 62 - 8

(n - 1)d = 54 ----(1)

S_{n} = 210

(n/2) [2a + (n - 1)d] = 210

By applying the values of a and (n - 1)d, we get

(n/2)[2(8) + 54] = 210

(n/2)[16 + 54] = 210

(n/2)(70) = 210

35 n = 210

Divide both sides by 35

n = 210/35 = 6

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

(6 - 1) d = 54

5d = 54

d = 54/5

Hence, the values of n and d are 6 and 54/5 respectively.

After having gone through the stuff given above, we hope that the students would have understood, how to find the missing term of an arithmetic sequence.

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