# How do you find the next three terms in 2,5,10,17,26,....?

Nov 10, 2015

Examine the sequence of differences of this sequence and its sequence of differences to find the next three terms, $37 , 50 , 65$ and the general formula ${a}_{n} = {n}^{2} + 1$

#### Explanation:

To find a pattern in this sequence, first write out the original sequence:

(i) $\textcolor{b l u e}{2} , 5 , 10 , 17 , 26$

Then write out the sequence of differences between successive terms of that sequence:

(ii) $\textcolor{b l u e}{3} , 5 , 7 , 9$

Then write out the sequence of differences of that sequence:

(iii) $\textcolor{b l u e}{2} , 2 , 2$

Having reached a constant sequence, there are a couple of things we can do:

(1) We can find the next three elements of the original sequence as requested.

To do this, add three more $2$'s to the last sequence:

$\textcolor{b l u e}{2} , 2 , 2 , \textcolor{red}{2} , \textcolor{red}{2} , \textcolor{red}{2}$

Then add three more terms to the previous sequence using the three new elements of this sequence as differences:

$\textcolor{b l u e}{3} , 5 , 7 , 9 , \textcolor{red}{11} , \textcolor{red}{13} , \textcolor{red}{15}$

Then add three more terms to the original sequence using the three new elements of this sequences as differences:

$\textcolor{b l u e}{2} , 5 , 10 , 17 , 26 , \textcolor{red}{37} , \textcolor{red}{50} , \textcolor{red}{65}$

(2) We can find a general formula for then $n$th term ${a}_{n}$ of the sequence by using the initial terms $\textcolor{b l u e}{2}$, $\textcolor{b l u e}{3}$, $\textcolor{b l u e}{2}$ of the sequences (i), (ii) and (iii) as coefficients:

a_n = color(blue)(2)/(0!) + color(blue)(3)/(1!) (n-1) + color(blue)(2)/(2!) (n-1)(n-2)

$= 2 + 3 n - 3 + {n}^{2} - 3 n + 2 = {n}^{2} + 1$

Apr 16, 2018

2," "5," "10," "17," "26," "color(blue)(37," "50," "65)

#### Explanation:

Looking at the sequence it should be clear that it is neither arithmetic nor geometric.

$2 , \text{ "5," "10," "17," } 26$

Look at the differences:

,$3 , \text{ "5," "7," } 9. \ldots .$

This pattern is a clue that squares are involved.

Look at the pattern of the differences:

$2 , \text{ "2," "2," "2..." } \leftarrow$ indicates that it is a pattern of squares.

${T}_{n} \rightarrow \text{ "2," "5," "10," "17," } 26$
$n \rightarrow \text{ "1," "2," "3," "4," } 5$
${n}^{2} \rightarrow \text{ "1," "4," "9," "16," } 25$

Comparing the square numbers with the given sequence, we can see that:

${T}_{n} = {n}^{2} + 1$

${T}_{6} = {6}^{2} + 1 = 37$

and so on to get the next terms as

2," "5," "10," "17," "26," "color(blue)(37," "50," "65)