# The third term in an arithmetic sequence is 10 and the fifth term is 26. If the first term is a 1, what is an equation for the nth term of this sequence?

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Jim G. Share
May 30, 2016

${a}_{n} = 8 n - 14$

#### Explanation:

The terms in the standard Arithmetic sequence are.

${a}_{1} , {a}_{1} + d , {a}_{1} + 2 d , {a}_{1} + 3 d , \ldots \ldots \ldots , {a}_{1} + \left(n - 1\right) d$

where ${a}_{1} \text{ is the 1st term and d, the common difference}$

${a}_{1} + \left(n - 1\right) d \text{ is the nth term}$

To obtain the nth term formula , we require to find ${a}_{1} \text{ and d}$

we are given: ${a}_{3} = 10 \text{ and } {a}_{5} = 26$

${a}_{3} = 10 \Rightarrow {a}_{1} + 2 d = 10 \Rightarrow {a}_{1} = 10 - 2 d \ldots . . \left(1\right)$

${a}_{5} = 26 \Rightarrow {a}_{1} + 4 d = 26 \Rightarrow {a}_{1} = 26 - 4 d \ldots \ldots . \left(2\right)$

equating (1) and (2): 10 -2d = 26 - 4d → d = 8

substitute d = 8 in (1) : ${a}_{1} = 10 - 16 = - 6$

We now have d = 8 and ${a}_{1} = - 6$

$\Rightarrow {a}_{n} = - 6 + 8 \left(n - 1\right) = 8 n - 14$

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May 17, 2017

Equation for nth term is ${a}_{n} = 8 n - 14.$

#### Explanation:

In this AP, the 3rd term is $10$ and first term is ${a}_{1}$

In an AP ${a}_{n} = {a}_{1} + \left(n - 1\right) d$

So by the formula: $\text{ } {a}_{3} = {a}_{1} + 2 d = 10$

where (d is common difference)

Similarly, ${a}_{5} = {a}_{1} + 4 d = 26$

Solving these two equations by subtracting: ${a}_{5} - {a}_{3}$
${a}_{1} + 4 d = 26$
$\underline{{a}_{1} + 2 d = 10}$
$\text{ } 2 d = 16$

we get $d = 8 \mathmr{and} {a}_{1} = - 6$

Now put these values in the formula:

a_n=a_1+(n-1)d" :

We get: ${a}_{n} = - 6 + \left(n - 1\right) 8$

${a}_{n} = 8 n - 14$

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