Let k = 1 n a k = n 2 + n . If a 10 = 59 a n d a 7 =7a1 , then is equal to [2026]

Question

Let $\sum_{k = 1}^{n} a_{k} = α n^{2} + β n .$ If $a_{10} = 59 and a_{6} = 7 a_{1}, then α + β$ is equal to [2026]

Smart Achievers · Accepted Answer

(4)

$a_{n} = S_{n} - S_{n - 1}$

$= (α n^{2} + β n) - (α {(n - 1)}^{2} + β (n - 1))$

$(1) a_{59} \Rightarrow 19 α + β = 59$

$(2) a_{6} = 7 a_{1} \Rightarrow 11 α + β = 7 (α + β)$

$\Rightarrow 2 α = 3 β$

$α = 3, β = 2$

$α + β = 5$

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