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Chapter 19: Problem 3

A sequence is defined by $$ x[k]=\frac{k^{2}}{2}+k, \quad k=0,1,2,3, \ldots $$ State the first five terms.

Short Answer

Expert verified
Answer: The first five terms of the given sequence are 0, \(\frac{3}{2}\), 4, \(\frac{15}{2}\), and 12.

Step by step solution

01

Understand the formula

We have a sequence defined by the following formula: $$ x[k]=\frac{k^{2}}{2}+k, \quad k=0,1,2,3, \ldots $$ We need to find the first five terms of this sequence by replacing k with consecutive natural numbers starting from 0.
02

Find the first term (k=0)

Plug in k = 0 into the formula: $$ x[0] = \frac{0^2}{2} + 0 = 0 $$ The first term of the sequence is 0.
03

Find the second term (k=1)

Plug in k = 1 into the formula: $$ x[1] = \frac{1^2}{2} + 1 = \frac{1}{2} + 1 = \frac{3}{2} $$ The second term of the sequence is \(\frac{3}{2}\).
04

Find the third term (k=2)

Plug in k = 2 into the formula: $$ x[2] = \frac{2^2}{2} + 2 = \frac{4}{2} + 2 = 2 + 2 = 4 $$ The third term of the sequence is 4.
05

Find the fourth term (k=3)

Plug in k = 3 into the formula: $$ x[3] = \frac{3^2}{2} + 3 = \frac{9}{2} + 3 = \frac{9}{2} + \frac{6}{2} = \frac{15}{2} $$ The fourth term of the sequence is \(\frac{15}{2}\).
06

Find the fifth term (k=4)

Plug in k = 4 into the formula: $$ x[4] = \frac{4^2}{2} + 4 = \frac{16}{2} + 4 = 8 + 4 = 12 $$ The fifth term of the sequence is 12. The first five terms of the given sequence are: 0, \(\frac{3}{2}\), 4, \(\frac{15}{2}\), and 12.

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Most popular questions from this chapter

A geometric sequence has first term 1 . The ninth term exceeds the fifth term by 240 . Find possible values for the eighth term.

(a) Obtain a quadratic Maclaurin polynomial approximation, \(p_{2}(x)\), to \(f(x)=\cos 2 x\). (b) Compare the approximate value given by \(p_{2}(1)\) with actual value \(f(1)\).

Obtain the Maclaurin series expansion for \(f(x)=\cosh x\).

Write down the 10th and 19 th terms of the arithmetic sequence (a) \(8,11,14, \ldots\) (b) \(8,5,2, \ldots\)

Find the 23 rd term of an arithmetic sequence with first term 2 and common difference \(7 .\)
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