Question

Find the sum of the geometric series given the following information

Answer 1

Answer:

-21844

Step-by-step explanation:

Given:

• $a_1=-4$
• $a_n=-16384$
• $r=4$

First find n by using the general form of a geometric sequence: $a_n=ar^{n-1}$   (where a is the first term and r is the common ratio)

$\implies -16384=(-4)(4)^{n-1}$

$\implies 4^{n-1}=\dfrac{-16384}{-4}$

$\implies 4^{n-1}=4096$

$\implies \ln 4^{n-1}=\ln 4096$

$\implies (n-1)\ln 4=\ln 4096$

$\implies n=\dfrac{\ln 4096}{\ln 4}+1$

$\implies n=6+1$

$\implies n=7$

Sum of the first n terms of a geometric series:

$S_n=\dfrac{a(1-r^n)}{1-r}$

(where a is the first term and r is the common ratio)

Substituting the given values and the found value of n into the formula:

$\implies S_{7}=\dfrac{(-4)(1-4^7)}{1-4}$

$\implies S_{7}=-21844$

Answer 2

Answer:

-21844

Step-by-step explanation:

Finding n

• aₙ = arⁿ⁻¹

• -16384 = -4(4)ⁿ⁻¹
• 4ⁿ⁻¹ = 4096
• 4ⁿ⁻¹ = 64²
• 4ⁿ⁻¹ = (8²)²
• 4ⁿ⁻¹ = (4³)²
• n - 1 = 6
• n = 7

Finding The Sum

• S₇ = -4(4⁷ - 1) / 4 - 1
• S₇ = -4 (16383) / 3
• S₇ = -65536/3
• S₇ = -21844