If x > 0 , and log 2 x + log 2 ( x ) + log 2 ( x 4 ) + log 2 ( x 8 ) + log 2 ( x 16 ) + … … = 4 , then find x .

Given, log 2 x + log 2 ( x ) + ( log 2 x 4 ) + log 2 ( x 8 ) + log 2 x 16 + . . . . . . . . . . . . = 4 ⇒ log 2 ( x . x . x 4 . x 8 . x 16 . . . . . . . . ) = 4 ⇒ log 2 ( x . x 1 2 . x 1 4 . x 1 8 . x 1 16 . . . . . . . . ) = 4 ⇒ log 2 ( x ( 1 + 1 2 + 1 4 + 1 8 + 1 16 + . . . . . . . . ) ) = 4 . . . . . . . . . i Now, 1 + 1 2 + 1 4 + 1 8 + 1 16 + . . . . . . . . . . are in G.P., whose 1 st term a = 1 & common ratio r = 1 2 As, sum of infinite terms in G.P. S ∞ = a 1 - r Therefore, S ∞ = 1 1 - 1 2 = 2 Therefore, from Equation i log 2 ( x 2 ) = 4 ⇒ x 2 = 2 4 ⇒ x 2 = 16 ⇒ x = ± 4 But x can't be negative ∴ x = 4 .

If x i > 0 , i = 1 , 2 , … , 50 and x 1 + x 2 + … + x 50 = 50 , then find the minimum value of 1 x 1 + 1 x 2 + … . . + 1 x 50 .

Given that x 1 + x 2 + … + x 50 = 50 , x i > 0 , i = 1 , 2 , . . . . , 50 We know that- For n positive real numbers, A.M ≥ H.M. x 1 + x 2 + x 3 + . . . . . + x 50 50 ≥ 50 1 x 1 + 1 x 2 + 1 x 3 + . . . . . + 1 x 50 ⇒ 50 50 ≥ 50 1 x 1 + 1 x 2 + 1 x 3 + . . . . . + 1 x 50 ⇒ 1 x 1 + 1 x 2 + 1 x 50 + 1 x 50 ≥ 50 Therefore, minimum value of 1 x 1 + 1 x 2 + 1 x 3 + , . . . . 1 x 50 is 50 .

E M B I B E

Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-2>Q 14

MEDIUM

JEE Main/Advance

IMPORTANT

Earn 100

A man arranges to pay off a debt of $Rs . 3600$ by $40$ annual installments which form an arithmetic series, when $30$ of the installments are paid he dies leaving a third of the debt unpaid, find the value of the first installment

Important Questions on Sequences and Series

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-2>Q 15

In a circle of radius

R

a square is inscribed, then a circle is inscribed in the square, a new square in the circle and so on for n times. If the ratio of the limit of the sum of areas of all the circles to the limit of the sum of areas of all the squares as

n \to \infty

is

k,

then find the value of

\frac{4 k}{π}

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-2>Q 16

If the common difference of the A.P. in which

T_{7} = 9

and

T_{1} T_{2} T_{7}

is least, is

' d'

then

20 d

is-

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-2>Q 17

If

x > 0,

and

\log_{2} x + \log_{2} (\sqrt{x}) + \log_{2} (\sqrt[4]{x}) + \log_{2} (\sqrt[8]{x}) + \log_{2} (\sqrt[16]{x}) + \dots \dots = 4,

then find

x .

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-2>Q 18

Given that

α, γ

are roots of the equation

A x^{2} - 4 x + 1 = 0

and

β, δ

the roots of the equation

B x^{2} - 6 x + 1 = 0,

then find value of

(A + B),

such that

α, β, γ & δ

are in H.P.

EASY

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-2>Q 19

If

x_{i} > 0, i = 1, 2, \dots, 50

and

x_{1} + x_{2} + \dots + x_{50} = 50,

then find the minimum value of

\frac{1}{x_{1}} + \frac{1}{x_{2}} + \dots . . + \frac{1}{x_{50}} .

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 5 - Sequences and Series>Exercise-2>Q 20

Find the sum of the infinitely decreasing G.P. whose third term, three times the product of the first and fourth term and second term form an A.P. in the indicated order, with a common difference equal to