## Archive for the ‘sequence’ Category

## Problem 797

Which of the following represents an example of the Fibonacci sequence in nature?

**Solution:-**

broccoli

## Problem 796

What is the first term of the Fibonacci sequence that is greater than 10?

Solution:-

13

## Problem 755

The government, through a subsidy program, distributes 31,000,000. If each person or agency spends 65% of what is received, and 65% of this is spend, and so on, how much total increase in spending results from this government action?

**Solution:-**

The total increase in spending is given by infinite series shown below.

31,000,000 + 31,000,000 (0.65) + 31,000,000(0.65)^{2} + …..

Thus , the series is geometric. When -1 < r < 1, the sum of an infinite geometric series is given by S_{∞} = , where r is the common ratio and a_{1} is the first term of the geometric series.

To find the common ratio, choose a term the sequence (other than the first) and divide by the preceding term.

r =

Since -1 < 0.65 < 1 , the series has a sum.

Use the formula to find the sum of the infinite geometric series. Substitute 0.65 for r and 31,000,000 for a_{1}.

S_{∞} =

=

Simplify

S_{∞} =

=

88,571,429

Therefore , the total effect on the economy will be approximately 88,571,429.

## Problem 741

Find the sum of the first 20 terms of the sequence: -1, 1, -1, 1, -1, …

**Solution:-**

Find a_{n} by using the explicit formula for geometric sequences: a_{n }= a_{1}r^{n-1} . Then use the formula for the sum of a finite geometric series:

S_{n} =

0.

## Problem 740

Write the geometric sequence that has four geometric means between 1and 1,024.

**Solution:-**

a_{n} and a_{1 } are given. Find the common ratio by using the explicit formula for geometric sequences: a_{n }= a_{1}r^{n-1} .

1, 4, 16, 64, 256, 1,024

## Problem 739

Find the next term of the sequence: -600, 300, -150, …

**Solution:-**

Find the common ratio and multiply the previous term by that value.

75

## Problem 738

Find the sum of the first 25 terms of the arithmetic series: -5, -4, -3, -2, -1, …

**Solution:-**

Find a_{n} by using the explicit formula for arithmetic sequences:

a_{n} = a_{1} + (n-1)d.

a_{1 }= -5

a_{25} = 19

The use the formula for the sum of an arithmetic series:

S_{n} = (a_{1} + a_{n})

175

## Problem 737

Find the sum of the first five terms of the arithmetic series:

a_{n} = 2n +2.

**Solution:-**

Find a_{1} and a_{5} and use the formula for the sum of an arithmetic series: S_{n} = (a_{1} + a_{n})

a_{1 }= 2* 1 + 2 = 4

a_{5} =2*5 +2 = 12

S_{n }= 40

## Problem 736

Find the 100^{th} term of the sequence: -4, -2, 0, 2, ….

**Solution:-**

Find the common difference and use the explicit formula for arithmetic sequences: .

a_{n} = a_{1} + (n-1)d.

a_{1 }= -2

n = 100

d = 2

194