Question

In a sequence of $21$ terms, the first $11$ terms are in AP with common difference $2$ and the last $11$ terms are in GP with common ratio $2$. If the middle term of an AP is equal to the middle term of the GP, then the middle term of the entire sequence is

• A. $-\frac{10}{31}$
• B. $\frac{10}{31}$
• C. $\frac{32}{31}$
• D. $-\frac{31}{32}$

Answer 1

The correct option is A

$-\frac{10}{31}$

Explanation for the correct option.

Step 1. Evaluate the AP terms.

The first $11$ terms are in AP.

Let the first term be $a$ and the common difference is $2$.

So the $11$th term of the AP is

$\begin{array}{rcl}{a}_{11}& =& a+(11-1)\times 2\\ & =& a+20\end{array}$

Now, the middle term of the AP is the sixth term, so the middle term is:

$\begin{array}{rcl}{a}_6& =& a+\left(6-1\right)\times 2\\ & =& a+10\end{array}$

Step 2. Evaluate the GP terms.

The next eleven terms in the sequence is in GP.

Let the first term of the GP be $b$and its common ratio is $2$.

So, the middle term of the GP series is:

$\begin{array}{rcl}{b}_6& =& b{\left(2\right)}^{6-1}\\ & =& 32b\end{array}$

Step 3. Form two equations.

The last term of the AP will be the same as the first term of the GP. So,

$a+20=b...\left(1\right)$

It is also given that the middle term of the AP is the same as the middle term of the GP that is $a_6=b_6$ and so

$a+10=32b...\left(2\right)$

Step 4. Solve for $a$ and $b$.

Using equation $1$ substitute $a+20$ for $b$ in equation $2$.

$$\begin{gathered} a+10=32\left(a+20\right) \\ \Rightarrow a+10=32a+640 \\ \Rightarrow -31a=630 \\ \Rightarrow a=-\frac{630}{31} \end{gathered}$$

Now, substitute $-\frac{630}{31}$ for $a$ in equation $1$.

$\begin{array}{rcl}b& =& -\frac{630}{31}+20\\ & =& \frac{-630+620}{31}\\ & =& -\frac{10}{31}\end{array}$

Step 5. Find the middle term of the sequence.

For the sequence of $21$ terms, the middle term is the $11$th term. So it is the last term of the AP or the first term of the GP.

The first term of the GP is $b=-\frac{10}{31}$.

So the middle term of the sequence is $-\frac{10}{31}$.

Hence, the correct option is A.