If the arithmetic mean and geometric mean of the $p^{\text {th }}$ and $q^{\text {th }}$ terms of the sequence $-16,8,-4,2, \ldots$ satisfy the equation $4 x^{2}-9 x+5=0$, then $p+q$ is equal to ______

JEE Main · 2021 · 26 Feb 2021 · Shift 2 · Sequences and Series

If the arithmetic mean and geometric mean of the

p^{th}

and

q^{th}

terms of the sequence

- 16, 8, - 4, 2, \dots

satisfy the equation

4 x^{2} - 9 x + 5 = 0

, then

p + q

is equal to ______

Answer: 10

The sequence

- 16, 8, - 4, 2, \dots

is geometric with first term

- 16

and ratio

- \frac{1}{2}

, so

a_{n} = - 16 (- \frac{1}{2})^{n - 1} = (- 1)^{n} 2^{5 - n}

The roots of

4 x^{2} - 9 x + 5 = 0

are

1

and

\frac{5}{4}

, and a geometric mean never exceeds the corresponding arithmetic mean, so

a_{p} a_{q} = 1

and

\frac{a _{p} + a _{q}}{2} = \frac{5}{4}

. Then

a_{p} a_{q} = 1

and

a_{p} + a_{q} = \frac{5}{2}

, so

a_{p}

and

a_{q}

are the roots of

t^{2} - \frac{5}{2} t + 1 = 0 \Rightarrow t = 2, \frac{1}{2}

From

a_{n} = (- 1)^{n} 2^{5 - n}

the value

2

occurs at

n = 4

and the value

\frac{1}{2}

n = 6

, both even indices as the positive signs require. Hence

p + q = 10

JEE Main · 2021 · 26 Feb 2021 · Shift 2 · Integer answer