k=1^n ak = n^2 + n. a10=59, a6=7a1. Find +. — Solution | JEE Mathematics

MCQMediumJEE 2026Arithmetic Progression (AP)

$\sum_{k=1}^n a_k = \alpha n^2 + \beta n$ . $a_{10}=59, a_6=7a_1$ . Find $\alpha+\beta$ .

Answer

Correct answer:B

Standard Method

Given:

S_n = \sum_{k=1}^n a_k = \alpha n^2 + \beta n

Also, $a_{10}=59$ and $a_6=7a_1$ .

Find: $\alpha+\beta$ .

Since $S_n$ is a quadratic expression in $n$ , the sequence $a_n$ is an arithmetic progression.

Using

a_n = S_n - S_{n-1}

we get

a_n = \alpha n^2 + \beta n - \left[\alpha(n-1)^2 + \beta(n-1)\right]

= \alpha(2n-1) + \beta

Now,

a_{10} = \alpha(19) + \beta = 59

19\alpha + \beta = 59

Also,

a_6 = 11\alpha + \beta

and

a_1 = \alpha + \beta

Given $a_6 = 7a_1$ , therefore

11\alpha + \beta = 7(\alpha + \beta)

11\alpha + \beta = 7\alpha + 7\beta

4\alpha = 6\beta

\beta = \frac{2}{3}\alpha

Substitute into $19\alpha + \beta = 59$ :

19\alpha + \frac{2}{3}\alpha = 59

\frac{59\alpha}{3} = 59

\alpha = 3

Hence,

\beta = 2

Therefore,

\alpha + \beta = 5

So, the correct option is B.

Use AP parameters directly

Given:

S_n = An^2 + Bn

with $a_{10}=59$ and $a_6=7a_1$ .

Find: $A+B$ .

For

S_n = An^2 + Bn

the terms form an AP with first term

a_1 = A+B

and common difference

d = 2A

So,

a_{10} = a_1 + 9d = (A+B) + 18A = 19A + B = 59

Also,

a_6 = a_1 + 5d = (A+B) + 10A = 11A + B

Given $a_6 = 7a_1$ ,

11A + B = 7(A+B)

4A = 6B

B = \frac{2}{3}A

Substituting into $19A+B=59$ gives

19A + \frac{2}{3}A = 59

A=3, \quad B=2

Hence,

A+B=5

So the correct option is B.

Assuming $a_n = \alpha n^2 + \beta n$ directly from the given sum. This is wrong because the expression is for $S_n$ , not for the term $a_n$ . First use $a_n = S_n - S_{n-1}$ .
Using the wrong expression for $a_n$ by forgetting to subtract $S_{n-1}$ carefully. Expanding $\alpha(n-1)^2$ incorrectly changes the coefficient of $n$ . Write the subtraction step fully before simplifying.
Applying $a_6 = 7a_1$ as $11\alpha+\beta = 7\alpha+\beta$ . This is wrong because $7a_1 = 7(\alpha+\beta)$ , so both terms inside the bracket must be multiplied by $7$ .

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