An AP consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the last three is 429. Then the third term in the AP is 11.

State true (T) or false (F).

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Detailed Solution

It is given that AP consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the last three is 429.
The expression to find the middle term is,

m = n + 1 2

.
So,

m = 37 + 1 2 m = 19

And, the sum of the three middlemost terms is 225 thus,

a 18 + a 19 + a 20 = 225

.... 1

.
Obtain the expression for the nth terms of an AP by using the formula,

a n = a + n − 1 d

a + 18 − 1 d + a + 19 − 1 d + a + 20 − 1 d = 225 a + 17 d + a + 18 d + a + 19 = 225 3 a + 18 d = 225 a + 18 d = 75 .... 2

And the sum of the last three is 429.

a 35 + a 36 + a 37 = 429 a + 35 − 1 d + a + 36 − 1 d + a + 37 − 1 d = 429 3 a + 105 d = 429 a + 35 d = 143 .... 3

Solve the equation (2) and (3),

a + 35 d = 143 a + 18 d = 75 − − − ¯ 17 d = 68 d = 4

So, putting value of d,

a + 354 = 143 a = 143 − 140 a = 3

Thus, the AP will be,

A P = a, a + d, a + 2 d, a + 3 d, .... A P = 3, 3 + 4, 3 + 24, 3 + 34, ... A P = 3, 7, 11, 15, ....

Therefore, the given statement, “An AP consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the last three is 429. Then the third term in the AP is 11.” is true.

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