The sum of $40$ terms of the series $1+2+3+4+5+8+7+16+9+......$ is,
- $398 + 2^{20}$
- $398 + 2^{21}$
- $398 + 2^{19}$
- $None\ of\ these$
The series is 1, 2, 3, 4, 5, 8, 7, 16, 9, 32... The odd terms are 1, 3, 5, 7, 9... (arithmetic progression). The even terms are 2, 4, 8, 16, 32... (geometric progression). Sum of 20 odd terms = 20^2 = 400. Sum of 20 even terms = 2(2^20 - 1) / (2 - 1) = 2^21 - 2. Total sum = 400 + 2^21 - 2 = 398 + 2^21.
Separate the given series into two alternating components: one arithmetic progression of odd numbers and one geometric progression of powers of 2. In 40 terms, there are 20 terms of the odd number progression, which sum to $20^2 = 400$. The 20 terms of the geometric progression sum to $2(2^{20} - 1) = 2^{21} - 2$. Adding these two partial sums gives $400 + 2^{21} - 2 = 398 + 2^{21}$. The result is $398 + 2^{21}$.