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

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5
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)

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

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#### Explanation

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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)

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