# How do you write the expression for the nth term of the sequence given 1, 4, 7, 10, 13,...?

Feb 27, 2017

$\therefore \text{nth term} = 1 + \left(n - 1\right) \cdot 3$

#### Explanation:

1;4;7;10;13;........... is an arithmetic series

${n}_{1} = 1$ and the common difference, $d = 3$

$\therefore \text{ nth term" = 1+(n-1)*3 }$

Mar 10, 2017

${T}_{n} = 3 n - 2$ is the expression for any term in the given sequence.

#### Explanation:

We should recognise that $1 , 4 , 7 , 10 , 13 \ldots .$ is an A.P. with a common difference of 3.

$d = 3$

The expression for the nth term can be determined from the formula, but there is a short cut.

T_n = d xx n + ???

As soon as see that $d = 3$, you will know that the expression for the nth term with start with ${T}_{n} = 3 n$

Now you just have find what must be added or subtracted to start at the correct first term.

Let $n = 1 \text{ "rarr 3 xx 1 = 3," but } {T}_{1}$ must be $1 ,$ so subtract 2

Let $n = 2 \text{ "rarr 3 xx 2= 6" but } {T}_{2}$ must be $4.$ so subtract 2

It would seem that the expression we want is: ${T}_{n} = 3 n - 2$

Check with ${T}_{3}$ which must be 7.

$3 \times 3 - 2 = 7$

${T}_{n} = 3 n - 2$

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Using the formula : ${T}_{n} = a + \left(n - 1\right) d$

In $1 , 4 , 7 , 10 , 14 , \ldots$

$a = {T}_{1} = 1 \text{ and } d = 3$

${T}_{n} = 1 + \left(n - 1\right) \times 3 \text{ } \leftarrow$ simplify

${T}_{n} = 1 + 3 n - 3$

${T}_{n} = 3 n - 2$