JEE Advanced · Mathematics · 5. Sequences & Series

Let $a_{1}, a_{2}, a_{3}, . . . . . . . . .$ be a sequence of positive integers in arithmetic progression with common difference $2$ . Also, let $b_{1}, b_{2}, b_{3}, . . . . . . . .$ be a sequence of positive integers in geometric progression with common ratio $2$ . If $a_{1} = b_{1} = c$ , then the number of all possible values of $c$ , for which the equality $2 (a_{1} + a_{2} + . . . . . . . . + a_{n}) = b_{1} + b_{2} + . . . . . + b_{n}$ holds for some positive integer $n$ , is _______

Verified Solution

Answer & Solution

Correct Answer

(A) 1

Step-by-step Solution

Detailed explanation

2 (a_{1} + a_{2} + . . . . . . . . + a_{n}) = b_{1} + b_{2} + . . . . . + b_{n}

\Rightarrow 2 \frac{n}{2} [2 a_{1} + (n - 1) \cdot 2] = \frac{b_{1} \cdot (2^{n} - 1)}{2 - 1}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Match List-I with List - II and select the correct answer using the code given below the lists

	List-I		List-II
A.	Volume of parallelepiped determined by vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $2$ . Then the volume of the parallelepiped determined by vectors $2 (\vec{a} \times \vec{b}), 3 (\vec{b} \times \vec{c})$ and $(\vec{c} \times \vec{a})$ is	P.	$100$
B.	Volume of parallel piped determined by vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $5$ . Then the volume of the parallelepiped determined by vectors $3 (\vec{a} + \vec{b}), (\vec{b} + \vec{c})$ and $2 (\vec{c} + \vec{a})$ is	Q.	$30$
C	Area of a triangle with adjacent sides determined by vectors $\vec{a}$ and $\vec{b}$ is $20$ .Then the area of the triangle with adjacent sides determined by vectors $(2 \vec{a} + 3 \vec{b})$ and $(\vec{a} - \vec{b})$ is	R.	$24$
D	Area of a parallelogram with adjacent sides determined by vectors $\vec{a}$ and $\vec{b}$ is $30$ . Then the area of the parallelogram with adjacent sides determined by vectors $(\vec{a} + \vec{b})$ and $\vec{a}$ is	S.	$60$

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