6 - Sequence and Series Questions Answers

Between two numbers, whose sum is 13/6, an even number of arithmetic means are inserted, the sum of these means exceed their number by unity. How many means are there?

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The expression ax2 + 2bx + b has same sign as that of b for every real x, then the roots of equation bx2 + (b – c) x + b – c – a = 0 are (A) real and equal (B) real and unequal (C) imaginary (D) None of these
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Find Sum to n terms

x/1-x²  +  x²/1-x⁴  +  x⁴/1-x⁸ + ....

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The sum of three positive real numbers x, y and z is 9. If the largest positive difference between any 2 of the 3 numbers is 5 and these 3 numbers are in arithmetic progression then the product of these 3 numbers will be what .please tell me.

Joshi sir comment

let the numbers are a-d, a, a+d

then according to the given condition    a-d+a+a+d = 9   so  3a = 9   so a = 3

and [a+d]-[a-d] = 5   so 2d = 5 so d =5/2

now solve

The AM of two positive numbers a & b exceeds their GM by 3/2 & the GM exceeds the HM by 6/5 such that a+b= α , |a-b|=β , then which of them are correct,

1) α + β² = 96            2) α + β² =74          3)α²+β = 234         4)α²+β = 84

Joshi sir comment

(a+b)/2 = √ab + 3/2                   (1)                        or         a+b = 2√ab + 3

√ab      = 2ab/(a+b)  +  6/5       (2)                        or         5(a+b)√ab  =  10ab + 6(a+b)

on solving these 2 we get               5[2√ab + 3]√ab = 10ab + 6[2√ab + 3]

or               10ab + 15√ab  =  10ab + 12√ab + 18

or                3√ab = 18     or      ab = 36          and      a+b = 15 = α

so a-b = √(225 - 144)     =    √81     =     9 = β