Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. (Assume that n begins with 1.) −5, 10 3 , − 20 9 , 40 27 , − 80 81 , ...

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11-03-2020
Mathematics

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Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. (Assume that n begins with 1.) −5, 10 3 , − 20 9 , 40 27 , − 80 81 , ...

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adamu4mohammed adamu4mohammed · Answer 1 · 2020-03-16T04:13:02+01:00

Answer:

The formula to the sequence

-5, 10/3, -20/9, 40/27, -80/81, ...

is

(-1)^n. 5×2^(n-1). 3^(1-n)

For n = 1, 2, 3, ...

Step-by-step explanation:

The sequence is

-5, 10/3, -20/9, 40/27, -80/81, ...

By inspection, we see the following

- The numbers are alternating between - and +

- The numerator of a number is twice the numerator of the preceding number. The first number is 5.

- The denominator of a number is 3 raised to the power of (1 minus the position of the number)

Using these, we can write a formula for the sequence.

(-1)^n for n = 1, 2, 3, ... takes care of the alternation between + and -

5×2^(n-1) for n = 1, 2, 3, ... takes care of the numerators 5, 10, 20, 40, ...

3^(1-n) for n = 1, 2, 3, ... takes care of the denominators 1, 3, 9, 27, ...

Combining these, we have the formula to be

(-1)^n. 5×2^(n-1). 3^(1-n)

For n = 1, 2, 3, ...

ashishdwivedilVT ashishdwivedilVT · Answer 2 · 2022-03-25T11:45:34+01:00

The formula for the general term of the sequence is

Tₙ = -5(-2/3)ⁿ where n=0,1,2,3.......

Given sequence is:

-5, 10/3, -20/9, 40/27......

The ratio of second to first term = -2/3

The ratio of second to the third term = -2/3

The ratio of the third to the fourth term = -2/3

Looking at the pattern, we can say that the given sequence is a Geometric Progression with the first term as -5 and the common ratio is -2/3.

What is the nth term of a GP?

The nth term of a GP with first term a and common ratio r is arⁿ.

So, the formula for the nth term for the given sequence will be:

Tₙ = -5(-2/3)ⁿ where n=0,1,2,3.......

Therefore, the formula for the general term of the sequence is

Tₙ = -5(-2/3)ⁿ where n=0,1,2,3.......

To get more about GP visit:

https://brainly.com/question/12006112