the sum of the 3rd and 7th terms of an A.P is 32 and their product is 220. find the sum of first 21th terms of A.P ( take value of D positive)
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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
Step-by-step explanation:
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