Find the first five terms of the sequence whose nth term is given. a: (-2)n + 2n \\b: n^2 - n...

Question:

Find the first five terms of the sequence whose nth term is given.

{eq}a: (-2)n + 2n \\b: n^2 - n \\c: |10 - n^2| {/eq}

Sequence

A sequence consists of array of terms that follows a general pattern. If the nth term of the sequence is given by the form {eq}a_n {/eq}, we can enumerate the terms of the sequence as {eq}a_1,a_2,a_3,a_4,a_5,... {/eq}, where {eq}a_1 {/eq} is the first term, {eq}a_2 {/eq} is the second term, {eq}a_3 {/eq} is the third term, and so on.

Answer and Explanation: 1

Using the concept above, we can list the first five terms as follows.

(a) {eq}-2n + 2n {/eq}

This can be simplified down into 0. So, the first five terms of this sequence are {eq}0,0,0,0,0 {/eq}.

(b) {eq}n^2 - n {/eq}

The first five terms can be obtained as follows.

$$\begin{align} a_1 &= (1)^{2} - 1\\[0.3cm] a_1 &= 1-1\\[0.3cm] a_1 &= 0\\[1.0cm] a_2 &= (2)^{2} -2\\[0.3cm] a_2 &= 4 - 2\\[0.3cm] a_2 &= 2\\[1.0cm] a_3 &= (3)^{2} -3\\[0.3cm] a_3 &= 9 - 3\\[0.3cm] a_3 &= 6\\[1.0cm] a_4 &= (4)^{2} -4\\[0.3cm] a_4 &= 16 - 4\\[0.3cm] a_4 &= 12\\[1.0cm] a_5 &= (5)^{2} -5\\[0.3cm] a_5 &= 25 - 5\\[0.3cm] a_5 &= 20\\[1.0cm] \end{align} $$

Hence, the first five terms are {eq}0,2,6,12,\ \text{and}\ 20 {/eq}.

(c) {eq}|10-n^{2}| {/eq}

$$\begin{align} a_1 &= |10-(1)^{2}|\\[0.3cm] a_1 &= |10-1|\\[0.3cm] a_1 &= 9\\[1.0cm] a_2 &= |10-(2)^{2}|\\[0.3cm] a_2 &= |10-4|\\[0.3cm] a_2 &= 6\\[1.0cm] a_3 &= |10-(3)^{2}|\\[0.3cm] a_3 &= |10-9|\\[0.3cm] a_3 &= 1\\[1.0cm] a_4 &= |10-(4)^{2}|\\[0.3cm] a_4 &= |10-16|\\[0.3cm] a_4 &= |-6|\\[0.3cm] a_4 &= 6\\[1.0cm] a_5 &= |10-(5)^{2}|\\[0.3cm] a_5 &= |10-25|\\[0.3cm] a_5 &= |-15|\\[0.3cm] a_5 &= 15\\[1.0cm] \end{align} $$

Hence, the first five terms of the sequence are {eq}9,6,1,6,\ \text{and}\ 15 {/eq}.


Learn more about this topic:

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Explicit Formula & Sequences: Definition & Examples

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Chapter 3 / Lesson 8
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What is an explicit formula for a sequence of numbers? Learn about the definition of explicit formula and how to find an explicit formula for arithmetic and geometric sequences, including examples.


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