Sequences And Series - Algebra - Mathematics Previous Year Questions

Given that x, y, and z are three consecutive positive integers and x \(-\) z + 2 = 0, what is the value of \({1 \over 2}{\log _e}x + {1 \over 2}{\log _e}z + {1 \over {2xz + 1}} + {1 \over 3}{\left( {{1 \over {2xz + 1}}} \right)^3} + ...\)?

The value of the sum \(\sum\limits_{k = 1}^\infty {\sum\limits_{n = 1}^\infty {{k \over {{2^{n + k}}}}} }\) is

If p, q, r are in AP and are positive, the roots of the quadratic equation px² + qx + r = 0 are all real for

If one GM, g and two AM's p and q are inserted between two numbers a and b, then (2p \(-\) q) (p \(-\) 2q) is equal to

If a₁, a₂, a₃, ......., a₂₀ are AM's between 13 and 67, then the maximum value of a₁, a₂, a₃, ......, a₂₀ is equal to

If a + 2b + 3c = 12, (a, b, c \(\in\)R⁺), then the maximum value of ab²c³ is

Sum of n terms of the infinite series

1.3² + 2.5² + 3.7² + ..... \(\infty\) is

Let a₁, a₂, a₃ .... be a harmonic progression with a₁ = 5 and a₂₀ = 25. The least positive integer n for which a_n < 0, is

Let a₁, a₂, ...... a₄₀ be in AP and h₁, h₂, ..... h₁₀ be in HP. If a₁ = h₁ = 2 and a₁₀ = h₁₀ = 3, then a₄h₇ is

In a sequence of 21 terms, the first 11 terms are in AP with common difference 2 and the last 11 terms are in GP with common ratio 2. If the middle term of AP be equal to the middle term of the GP, then the middle term of the entire sequence is

Let \(\frac{1}{16}, a\) and \(b\) be in GP and \(\frac{1}{a}, \frac{1}{b}, 6\) be in AP, where \(a, b>0\). Then, \(72(a+b)\) is equal to

Given, a sequence of 4 numbers, first three of which are in GP and the last three are in AP with common difference 6. If first and last term of this sequence are equal, then the last term is

If \(a_1, a_2, \ldots, a_n\) are in HP, then the expression \(a_1 a_2+a_2 a_3+\ldots+a_{n-1} a_n\) is equal to

The coefficient of $x^n$ in the expansion of $\frac{1-a x-x^2}{e^x}$ is

If $a, b, c, d$ be four positive unequal quantities and $s=a+b+c+d$, then $(s-a)(s-b)(s-c) (s-d)>k a b c d$. Then, value of $k$ is

For three numbers $a, b, c$ between 2 and 18 such that their sum is 25 , the numbers $2, a, b$ are in AP and the numbers $b, c, 18$ are in GP Then, the value of $a+b+c$ is