An A P consists of 12 terms whose sum is 354 The ratio of the sum of the even - Maths - Sequences and Series

Question

An A P consists of 12 terms whose sum is 354 The ratio of the
sum of the even terms to the sum of the odd terms is 32:27 Find the
common difference of the progression

Ajanta Trivedi · Accepted Answer

let the first term and the common difference of the AP be a and d respectively.

sum of 12 terms = 354

$\frac{12}{2} [2 a + (12 - 1) d] = 354 6 [2 a + 11 d] = 354 2 a + 11 d = 59............ (1)$

let the sum of even terms = 32x and the sum of odd terms = 27x.

therefore

$32 x + 27 x = 354 59 x = 354 x = \frac{354}{59} = 6$

sum of odd terms = 27*6 = 162

odd terms are $t_{1}, t_{3}, t_{5}, t_{7}, t_{9}, t_{11}$ .

$t_{1} = a a n d t_{3} - t_{1} = a + 2 d - a = 2 d$

the common difference for the series containing odd number of terms = 2d.

the number of odd terms = 6

thus the sum of the odd terms

$\frac{6}{2} [2 a + (6 - 1) * 2 d] = 162 3 [2 a + 10 d] = 162 6 [a + 5 d] = 162 a + 5 d = \frac{162}{6} a + 5 d = 27 ............ (2)$

multiplying eq(2) by 2 and subtracting by eq(1)

$2 a + 11 d = 59 2 a + 10 d = 54 d = 5$

thus the common difference of the progression is 5.

hope this helps you.

cheers!!

An A.P consists of 12 terms whose sum is 354. The ratio of the sum of the even terms to the sum of the odd terms is 32:27. Find the common difference of the progression.