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Minchanka [31]
2 years ago
14

Find the sum of the geometric series given the following information

Mathematics
2 answers:
BaLLatris [955]2 years ago
5 0

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

san4es73 [151]2 years ago
3 0

Answer:

<u>-21844</u>

Step-by-step explanation:

<u>Finding n</u>

  • 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

<u>Finding The Sum</u>

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