Question

\Last term of a geometric sequence is 4096 and the first term is 8. What are the two geometric means?



Write the four geometric means between 10 and 10000.


1. Sn = 244, r = -3, n = 5 a1 =?
2. an = 324, n = 3, Sn = 484, a1 =?
3. Sn = 1022, r = 2, n = 9, a1 =?



Find the sum of each infinite geometric sequence.
2, 6, 18, 54, …
a1 = 1, r = 1/2
a1 = -8, r = -3/4

Answer 1

Answer:

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Step-by-step explanation:

Hint: Here in this question, we have to find the sum of finite geometric series. The geometric series is defined as the series with a constant ratio between the two successive terms. Then by considering the geometric series we have found the sum of the series.

Complete step-by-step solution:

In mathematics we have three types of series namely, arithmetic series, geometric series and harmonic series. The geometric series is defined as the series with a constant ratio between the two successive terms. The finite geometric series is generally represented as a,ar,ar2,...,arn

, where a is first term and r is a common ratio.

Now consider the series 4096–512+64−…..

Here the term a is known as first term. the value of a is 4096.

The r is the common ratio of the series. It is defined as r=a2a1

The value of r is determined by r=−5124096=−18

Now we have to find the sum of finite geometric series, the sum for finite geometric series is defined by Sn

Here the value of r is less than 1 we have a formula for the sum of geometric series and it is defined as

Sn=a(1−rn)(1−r)

Here the value of n is 5.

Therefore by substituting the values in the formula we have

S5=4096(1−(−18)5)1−(−18)

On simplifying we have

S5=4096(1+(18)5)1+18

⇒S5=4096(1+132768)8+18⇒S5=4096(32768+132768)8+18⇒S5=4096(3276932768)98⇒S5=4096×3276932768×89⇒S5=3641

Hence the sum of geometric series 4096–512+64−…to5 terms is 3641.

Note: Three different forms of series are arithmetic series, geometric series and harmonic series. For the arithmetic series is the series with common differences. The geometric series is the series with a common ratio. The sum is known as the total value of the given series.