Question

Toro
(1) The sum of the 3rd and 7th terms of an AP is 32 and their product is 220. Find
the sum of first twenty-one terms of the A.P. Make the value of d positive).​

Answer 1

Step-by-step explanation:

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

Case I : The sum of the third and seventh term of an AP is 32.

=> a + 2d + a + 6d = 32

=> 2a + 8d = 32

=> 2(a + 4d) = 32

=> a + 4d = 16

=> a = 16 - 4d ______(i)

Case II : Their product is 220.

=> (a + 2d) (a + 6d) = 220

=> (16 - 4d + 2d) (16 - 4d + 6d) = 220 [from equation (i)]

=> (16 - 2d) (16 + 2d) = 220

=> (16)² - (2d)² = 220

=> 256 - 4d² = 220

=> 4d² = 256 - 220

=> d² = 36/4

=> d = √9 = 3

Substituting the value of d in equation (i) :

a = 16 - 4 × 3 = 4

Sum of first twenty one term = n/2 [2a + (n - 1)d]

= 21/2 [ 2 × 4 + (21 - 1)3]

= 21/2 [ 8 + 60]

= (21 × 68)/2

= 714

Hence,

Sum of first twenty one term = 714