Question

Study the given series and answer the questions: 64 +96 + 144 + ....

(i) If the sum of the terms of the series is 844, how many terms are there?
(ii) What is the last term of the series? ​

Answer 1

(i) 5

(ii) 324

Explanation:

(i)

64 + 96 + 144 + ... a_n = 844

r = 96/64 = 144/96 = 1.5

Sum of geometric series:

S_n = a_1(1 - r^n)/(1 - r)

64(1 - 1.5^n)/(1 - 1.5) = 844

-128 × (1 - 1.5^n) = 844

1 - 1.5^n = -844/128

1 - 1.5^n = -211/32

-1.5^n = -243/32

1.5^n = 243/32

1.5^n = 3^5/2^5

1.5^n = (3/2)^n

1.5^n = 1.5^5

n = 5

(ii) a_n = a_1 × r^(n - 1)

a_5 = 64 × r^(5 - 1)

a_5 = 64 × 1.5^4

a_5 = 324