JAMIA 2022 — Mathematics PYQ

1. Identify the Terms

The sequence has 21 terms: $a_1,a_2,\dots ,a_{21}$.

• The A.P. Part (Terms 1 to 11):

  Common difference $d=2$.

  The middle term of this A.P. is the $6^{th}$ term ($a_6$).

• The G.P. Part (Terms 11 to 21):

  Common ratio $r=2$.

  The middle term of this G.P. is the $16^{th}$ term ($a_{16}$).

• The Entire Sequence (Terms 1 to 21):

  The middle term of the entire sequence is the $11^{th}$ term ($a_{11}$).

2. Set Up the Equations

Let the $11^{th}$ term (the junction point) be $x$.

• From the A.P. (going backwards from $a_{11}$ to $a_6$):

  $$a_6=a_{11}-5d$$
  $$a_6=x-5(2)=x-10$$

• From the G.P. (going forwards from $a_{11}$ to $a_{16}$):

  $$a_{16}=a_{11}\cdot r^{(16-11)}$$
  $$a_{16}=x\cdot 2^5=32x$$

3. Use the Given Condition

The problem states that the middle term of the A.P. is equal to the middle term of the G.P.:

Substitute the expressions we found:

4. Solve for $x$

Rearrange the equation to solve for $x$:

Since $x$ represents $a_{11}$, which is the middle term of the entire 21-term sequence, we have our answer.

Final Answer

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