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murzikaleks [220]
2 years ago
11

Can some one help me with geometric sequence?​

Mathematics
1 answer:
Yuliya22 [10]2 years ago
8 0

Answer:

See below

Step-by-step explanation:

We have the series

$\sum _{ n=0 }^{ \infty }{ 3(0.8)^n}$

(a) To show that the sum of first n terms is S_n= 15(1-0.8^n)

Consider that the this sum is given as

S_n = 3\dfrac { 1-{ r }^n}{ 1-r }

In this case, the ratio is 0.8, thus, r=0.8

S_n = 3\dfrac { 1-{ 0.8 }^n }{ 1-0.8 }

3\dfrac { 1-{ 0.8 }^{ n } }{ 1-0.8 } = 15(1-0.8^n) \iff  3(1-{ 0.8 }^{ n }) = 15(1-0.8^n)( 1-0.8) \iff  3 = 15(1-0.8) \iff 3 = 15-12 = 3

Therefore,

S_n = 3\dfrac { 1-{ 0.8 }^n }{ 1-0.8 } = 15(1-0.8^n)

\dfrac{S_6}{S_4} = \dfrac{15(1-0.8^6)}{15(1-0.8^4)} = \dfrac{(1-0.8^6)}{(1-0.8^4)} = \dfrac{0.737856}{0.5904} = \dfrac{737856}{5904} =  \dfrac{144\cdot 5124}{144\cdot 41} = \boxed{\dfrac{5124}{41} }

(b) Calculate the smallest integer value of n such that S_{40} -S_n   < \dfrac{1}{2}

Once S_{40} \approx 15 \text{ and } S_n>14.5, \text{ thus }  S_{40} \approx S_n, \text{ but } n \text{ is considerably smaller than 40}

I am considering S_{40}=15, so

15-15(1-0.8^n) < \dfrac{1}{2}  \implies  0.8^n < 0.03 \implies n> 15.71

\text{Once } n\in\mathbb{Z}, \text{ then }  n=16

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