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Let an denote the nth term of a geometric sequence. let a0=256 and an+1=0.5an .

part a

write an explicit formula for this sequence.

an=

part b

to the nearest whole number, what is the sum of the first 25 terms?

Sagot :

MrRoyal

The explicit formula is [tex]a_{n-1} = 256 * 0.5^{n-1[/tex] and the sum of the first 25 terms is approximately 512

The explicit formula

The given parameters are:

[tex]a_0 = 256[/tex]

[tex]a_{n +1} = 0.5a_n[/tex]

Set n = 0

[tex]a_1 = 0.5a_0[/tex]

Substitute 256 for a0

[tex]a_1 = 0.5 * 256[/tex]

[tex]a_1 = 128[/tex]

This means that the common ratio is 0.5.

The explicit formula is then represented as:

[tex]a_{n-1} = a_0 * r^{n-1[/tex]

This gives

[tex]a_{n-1} = 256 * 0.5^{n-1[/tex]

The sum of the first 25 terms

This is calculated using:

[tex]S_n = \frac{a_0(r^{n-1} -1)}{r - 1}[/tex]

This gives

[tex]S_{25} = \frac{256 * (0.5^{25-1} -1)}{0.5 - 1}[/tex]

Evaluate

[tex]S_{25} = 512[/tex]

Hence, the sum of the first 25 terms is approximately 512

Read more about geometric series at:

https://brainly.com/question/24643676

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