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.

the answer is simply just 6n add the two together as they have an n

c. 27 units.

step-by-step explanation:

we are told that in circle a, ∠bae ≅ ∠dae.

we can see from our given diagram that in and ;

, as these are radii of our given circle.

therefore, by sas congruence postulate.

hence, side be will be equal to side de as corresponding parts of congruent triangles are congruent. so we can set an equation to find the value of x as:

now let us substitute x=17 in the expression for the length of be.

therefore, length of be will be 27 units and option c is the correct choice.