JEE Advanced 2020 · Paper 1 · Mathematics Q14

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MathematicsNumerical+4Question 50 of 54

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive integers in arithmetic progression with common difference 2. Also, let $b_1, b_2, b_3, \ldots$ 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

$2(a_1 + a_2 + \cdots + a_n) = b_1 + b_2 + \cdots + b_n$

holds for some positive integer $n$ , is ______

Your answer

Answer and worked solution

Official answer: 1

The equation.
$2\cdot\frac n2\left[2c+2(n-1)\right]=c(2^n-1)\implies c\,(2^n-1-2n)=2n(n-1).$

Small $n$ .

$n=1$ : $c=0$ , not positive.
$n=2$ : $c=-4$ , not positive.
$n=3$ : $c=\frac{12}{1}=12$ . ✓
$n=4,5,6$ : $c=\frac{24}7,\frac{40}{21},\frac{60}{51}$ , not integers.

Large $n$ . For $n\ge7$ , $2^n-1-2n>2n(n-1)$ , so $0<c<1$ .

Only $c=12$ works. Answer: 1

Topics:Arithmetic progression Geometric progression

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JEE Advanced 2020 · Paper 1, Mathematics question 14