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

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

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Answer 1 · 2015-11-10T18:34:23

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

Explanation:

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

(i) $2, 5, 10, 17, 26$ $color(blue)(2),5,10,17,26$

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

(ii) $3, 5, 7, 9$ $color(blue)(3),5,7,9$

Then write out the sequence of differences of that sequence:

(iii) $2, 2, 2$ $color(blue)(2),2,2$

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 $2$ $2$ 's to the last sequence:

$2, 2, 2, 2, 2, 2$ $color(blue)(2),2,2,color(red)(2),color(red)(2),color(red)(2)$

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

$3, 5, 7, 9, 11, 13, 15$ $color(blue)(3),5,7,9,color(red)(11),color(red)(13),color(red)(15)$

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

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

(2) We can find a general formula for then $n$ $n$ th term $a_{n}$ $a_n$ of the sequence by using the initial terms $2$ $color(blue)(2)$ , $3$ $color(blue)(3)$ , $2$ $color(blue)(2)$ of the sequences (i), (ii) and (iii) as coefficients:

$a_{n} = \frac{2}{0!} + \frac{3}{1!} (n - 1) + \frac{2}{2!} (n - 1) (n - 2)$ $a_n = color(blue)(2)/(0!) + color(blue)(3)/(1!) (n-1) + color(blue)(2)/(2!) (n-1)(n-2)$

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

Answer 2 · 2018-04-16T08:02:24

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

Explanation:

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

$2, 5, 10, 17, 26$ $2," "5," "10," "17," "26$

Look at the differences:

, $3," "5," "7," "9.....$

This pattern is a clue that squares are involved.

Look at the pattern of the differences:

$2," "2," "2," "2..." "larr$ indicates that it is a pattern of squares.

$T_n rarr" "2," "5," "10," "17," "26$
$nrarr" "1," "2," "3," "4," "5$
$n^2rarr" "1," "4," "9," "16," "25$

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

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