Animated Solution for Mathematics - Sequence and Series: Let ${\sum }_{k=1}^n{a}_k=\alpha n^2+\beta n$. If $a_{10}=59$ and $a_6=7a_1$, then $\alpha +\beta$ is equal to

We are given the sum of $n$ terms as $S_n=\alpha n^2+\beta n$. Note that this expression is a quadratic lacking a constant term, which is the mathematical signature of an Arithmetic Progression (A.P.).

Recognizing this structure is the first step in decoding the sequence. It confirms that the common difference is constant and the sequence behaves linearly.

To find the general term $a_n$, we utilize the fundamental relation:

Imagine a stack of $n$ blocks; removing the stack of $n-1$ blocks leaves exactly the $n$-th block. This logic holds firm for all $n\ge 2$. For the first term, $a_1$, we simply evaluate $S_1$.

We substitute $S_n$ and $S_{n-1}$ into our bridge equation:

By expanding the terms and grouping the coefficients of $\alpha$ and $\beta$, we simplify the expression to:

This is our master key. It allows us to calculate any term in the sequence provided we determine the values of $\alpha$ and $\beta$.

We apply the given constraints: $a_{10}=59$ and $a_6=7a_1$. Substituting $n=10$ into our master key yields:

Next, we express $a_6$ and $a_1$ using the master key:

The condition $a_6=7a_1$ transforms into:

Simplifying this equation, we find $4\alpha =6\beta$, which leads to:

We substitute $\beta =\frac{2\alpha }{3}$ back into our first equation ($19\alpha +\beta =59$):

Multiplying the entire equation by $3$ gives $57\alpha +2\alpha =177$, which simplifies to $59\alpha =177$. Thus, we find:

Substituting $\alpha =3$ back into our expression for $\beta$, we get $\beta =2$. The final result is: