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 = \frac{a_{1}}{1-r}, where r is the common ratio and a1 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 = \frac{31,000,000(0.65)}{31,000,000}=0.65

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 a1.

S = \frac{a_{1}}{1-r}

= \frac{31,000,000}{1-0.65}

Simplify

S = \frac{31,000,000}{1-0.65}

=\frac{31,000,000}{0.35}

\approx 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  an  by using the explicit formula for geometric sequences: an = a1rn-1   . Then use the formula for the sum of a finite geometric series:

Sn =  \frac{a_{1}-a_{1}r^{n}}{1-r}

0.

 

 

Problem 740

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

 

Solution:-

 

an and  a1  are given. Find the common ratio by using the explicit formula for geometric sequences:  an = a1rn-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 an by using the explicit formula for arithmetic sequences:

an = a1 + (n-1)d.

a1 = -5

a25 = 19

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

Sn = \frac{n}{2}(a1 + an)

175

 

Problem 737

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

an = 2n +2.

 

Solution:-

 

Find a1 and  a5  and use the formula for the sum of an arithmetic series:    Sn = \frac{n}{2} (a1 + an)

a1 = 2* 1 + 2 = 4

a5 =2*5 +2 = 12

Sn = 40

 

Problem 736

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

 

Solution:-

 

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

an = a1 + (n-1)d.

a1  = -2

n = 100

d = 2

194