# What is the nth term of 2,6,11,17,24? Thanks

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#### Explanation

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#### Explanation:

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Mar 8, 2018

${a}_{n} = \frac{1}{2} \left({n}^{2} + 5 n - 2\right)$

#### Explanation:

We can identify this quadratic sequence by the method of differences:

Write down the given sequence:

$\textcolor{b l u e}{2} , 6 , 11 , 17 , 24$

Write down the sequence of differences between consecutive terms:

$\textcolor{b l u e}{4} , 5 , 6 , 7$

Write down the sequence of differences of those differences:

$\textcolor{b l u e}{1} , 1 , 1$

Having reached a constant sequence, we can match the original sequence with a formula using the initial term of each of these sequences as coefficients:

a_n = color(blue)(2)/(0!)+color(blue)(4)/(1!)(n-1)+color(blue)(1)/(2!)(n-1)(n-2)

$\textcolor{w h i t e}{{a}_{n}} = 2 + 4 n - 4 + \frac{1}{2} {n}^{2} - \frac{3}{2} n + 1$

$\textcolor{w h i t e}{{a}_{n}} = \frac{1}{2} \left({n}^{2} + 5 n - 2\right)$

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