The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of four consecutive terms of it, prove that the resulting sum is square of an integer.
My Self Assessment
The angles of a triangle are in Arithmetic Progression and let . Find the angle A.
If are in Arithmetic Progression where for all i, show that
Find three numbers a, b, c between 2 and 18 such that (i) their sum is 25 (ii) 2, a, b are consecutive terms of an Arithmetic Progression and (iii) the numbers b, c, 18 are consecutive terms of a Geometric Progression
If n is a natural number such that and are distinct primes, then show that
The sum of the squares of three distinct real numbers which are in Geometric Progression is . If their sum is , show that
If are in Arithmetic Progression, determine the value of x.
If a, b, c are in Arithmetic Progression and are in Harmonic Progression then prove that either or a, b and are in Geometric Progression.
If a, b, c are positive real numbers then prove that
Let the Harmonic Mean and Geometric Mean of two positive numbers be in the ratio of 4:5. Then the two numbers are in the ratio . . . . .
Let p, q be the roots of the equation , and r and s are roots of the equation . If are in arithmetic progression then A = . . . . . , B = . . . . .
Let x be the Arithmetic Mean and y, z be two Geometric Means between any two positive numbers then
Given positive integers r > 1, n > 2 and the coefficients of (3r)th term and (r + 2)th terms in the binomial expansion of (1 + x)2n are equal then
a) n = 2r
b) n = 2r + 1
c) n = 3r
d) none of these
Coefficient of x4 in is
a)
b)
c)
d) None of these
The product of n positive real numbers is unity. Then their sum is
a) A positive integer
b) Divisible by n
c) Equal to
d) Never less than n
If in the expansion of (1 + x)m (1 – x)n, the coefficients of x and x2 are 3 and –6 respectively. then m is
a) 6
b) 9
c) 12
d) 24
In the binomial expansion of the sum of the 5th term and 6th term is zero, then equals
d)
Coefficient of t24 in (1 + t2)12 (1 + t12) (1 + t24) is
Prove that is divisible by 25 for any natural number n.
Use mathematical induction to prove that is divisible by 24 for all n > 0.
Find the sum of the series
Prove that C0 – 22C1 + 32C2 − . . . + (−)n (n + 1)2 Cn = 0 for n > 2 where
Find the natural number a for which where the function f satisfies the relation f (x + y) = f (x).f(y)for all natural numbers x and y and further f (1) = 2
Using mathematical induction prove that
