Find the middle term of the A.P. 213, 205, 197,…, 37

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Find the middle term of the $A.P. 213, 205, 197, \dots, 37$

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125

128

130

120

answer is A.

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Given arithmetic progression,

213, 205, 197, \dots, 37

.
In this progression,
First term a = 213, second term = 205 and last term

a_{n} = 37

Then common difference,
d = second term – first term
d = 205 – 213
d = -8
Suppose number of terms = n
Thus, the last term formula of the series is,

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

37 = 213 + (n - 1) (- 8)

8 n = 184

n = 23

So, the middle terms is,

= \frac{n + 1}{2}

= \frac{23 + 1}{2}

= \frac{24}{2}

= 12

Hence, the middle term = 12th So, the 12th term is,

T_{12} = a + 11 d

T_{12} = 213 + 11 (- 8)

T_{12} = 125

Thus, the middle term of progression = 125.
Hence, the correct option is 1.

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