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Sequence
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Sequence

by Reynd Gunsin
(Kota Kinabalu,Sabah, Malaysia)











































Given a sequence: -2, -1, -1/2, -1/4,..., -1/512

(i) Find the number of terms.

(ii) Find the sum of all terms in the sequence.

Comments for Sequence

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Jul 22, 2011
Sequence
by: Staff


The question:

by Reynd Gunsin
(Kota Kinabalu,Sabah, Malaysia)


Given a sequence: -2, -1, -1/2, -1/4,..., -1/512

(i) Find the number of terms.

(ii) Find the sum of all terms in the sequence.


The answer:


-2, -1, -1/2, -1/4,..., -1/512 is a geometric sequence. It has the (general) form:

a_n = a_1 * (r)^(n - 1)


a_n = x with a subscript of n (this is the nth term in the sequence)

a_1 = x with a subscript of 1 (this is the 1st term in the sequence)

n = number of terms

r = the common ratio

r, the common ratio, can be calculated as follows:
r_n = a_n/a_n-1 (n must be greater than 1)

r_n = r with a subscript of n (this is the common ratio)

a_n = x with a subscript of n (this is the nth term in the sequence)

a_n-1 = x with a subscript of n-1 (this is the n-1 term in the sequence)

r_3 = a_3/a_2

r_3 = (-1/2)/ (-1)

The common ratio, r = 1/2

The n-th term of the geometric sequence is:

a_n = a_1 * (r)^(n - 1)

a_1 = -2

r = ½


a_n = (-2) * (1/2)^(n - 1)



(i) Find the number of terms by solving for n.

a_n = (-2) * (1/2)^(n - 1)


the last term in the series is: -1/512


-1/512 = (-2) * (1/2)^(n - 1)

-1/512 = (-2) * [(1)^(n - 1)]/[(2)^(n - 1)]

-1/512 = (-2) * 1/[(2)^(n - 1)]

-1/512 = (-2)/[(2)^(n - 1)]


Multiply each side of the equation by -1

(-1)*(-1/512) = (-1)*(-2)/[(2)^(n - 1)]

1/512 = 2/[(2)^(n - 1)]


Multiply each side of the equation by 512

512 * (1/512) = 512 * 2/[(2)^(n - 1)]

1 = 512 * 2/[(2)^(n - 1)]

1 = 1024/[(2)^(n - 1)]


Multiply each side of the equation by (2)^(n - 1)

[(2)^(n - 1)] * 1 = [(2)^(n - 1)] * 1024/[(2)^(n - 1)]

2^(n - 1) = [(2)^(n - 1)] * 1024/[(2)^(n - 1)]

2^(n - 1) = 1024 * [(2)^(n - 1)/(2)^(n - 1)]

2^(n - 1) = 1024 * 1

2^(n - 1) = 1024

2^(n - 1) = 2^10


Compute the log base(2) of each side of the equation

2^(n - 1) = 2^10

log_2 [2^(n - 1)] = log_2 [2^10]

n - 1 = 10

solve for n

n - 1 + 1 = 10 + 1

n + 0 = 10 + 1

n = 10 + 1

n = 11, there are 11 terms in the sequence



we can check this solution using the original equation to compute the 11th term

a_n = (-2) * (1/2)^(n - 1)

a_11 = (-2) * (1/2)^(11 - 1)

a_11 = (-2) * (1/2)^(10)

a_11 = -1/512, OK



(ii) Find the sum of all terms in the sequence.
The sum of the terms in the sequence is called the geometric series.

In this case, the sequence does not go on forever (an infinite number of terms). There are only n terms in the sequence.

The nth partial sum of a geometric series can be computed using the following formula:

S_n = a_1 * (1-rn) /(1-r)

a_1 = -2

r = ½

n = 11

S_11 = (-2) * [1-(½)¹¹]/(1-½)


S_11 = (-2) * [1- .000488281]/(0.5)]

S_11 = (-2) * [0.999512]/(0.5)]

S_11 = (-2) * 1.99902

S_11 = -3.99805


The final answer to your question is:


(i) number of terms, n = 11

(ii) sum of all 11 terms in the sequence, S_11 = -3.99805






Thanks for writing.

Staff
www.solving-math-problems.com



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