Find the sum of the geometric series 512+256+...+4
a) 510
b) 1,022
c) 1,020
d) 1,016

Option C. 1020

Step-by-step explanation:

From the question given above,

512 + 256 +... + 4 =?

We'll begin by calculating the number of terms in the sequence. This can be obtained as follow:

First term (a) = 512

Common ratio (r) = 2nd term / 1st term

Common ratio (r) = 256 /512

Common ratio (r) = 1/2

Last term (L) = 4

Number of term (n) =?

Tₙ = arⁿ¯¹

L = arⁿ¯¹

4 = 512 × (1/2)ⁿ¯¹

Divide both side by 512

4 / 512 = (1/2)ⁿ¯¹

1/128 = (1/2)ⁿ¯¹

Express 128 in index form with 2 as the base

1/2⁷ = (1/2)ⁿ¯¹

(1/2)⁷ = (1/2)ⁿ¯¹

Cancel 1/2 from both side

7 = n – 1

Collect like terms

7 + 1 = n

n = 8

Thus, the number of terms is 8

Finally, we shall determine the sum of the series as follow:

First term (a) = 512

Common ratio (r) = 1/2

Number of term (n) = 8

Sum of 8th term (S₈) = ?

Sₙ = a[1 – rⁿ] / 1 – r

S₈ = 512 [1 – (½)⁸] / 1 – ½

S₈ = 512 [1 – 1/256] ÷ ½

S₈ = 512 [255/256] × 2

S₈ = 2 × 255 × 2

S₈ = 1020

Thus, the sum of the series is 1020.