The sum$1 + 3 + 11 + 25 + 45 + 71+..$upto 20 terms, is equal to
Step-by-Step Solution
Key Concept: Fit the sequence$1,3,11,25,45,71,\ldots$to a$quadratic-difference$pattern and sum up to$20$terms.
Given sum is$(1)$$S_n = 1 + 3 + 11 + 25 + 45 + 71$+$\$ldots + Tn$First order differences are i_n A.P. Thus, we can assume that 2$Tn = an + bn$+ c$⎧ ⎪$T_{1} = 1 = a + b + c$⎫ ⎪ Solving ⎨$T = 3 = 4a + 2b + c$⎬,2 ⎩ ⎪ ⎭ ⎪$T_{3} = 11 = 9a + 3b + c$we get$a = 3$, b = -7,$c = 5$Hence, general term of given series is 2$T_n = 3n - 7n + 5$Hence, required sum equals$n=20$20$\cdot21$$\cdot41 20$$\cdot21 2$$\sum ($3n - 7n + 5) = 3$( ) - 7( ) + 5$(20)$= 7240$n=1$6 2$
Correct Answer: 1