Sequence of series practice · Medium

If $a_1=a_2=2, a_n=a_{n-1}-1(n>2)$, then $a_5$ is

The sum of $\frac{\frac{1}{2}\cdot \frac{2}{2}}{1^3}+\frac{\frac{2}{2} \cdot \frac{3}{2}}{1^3+2^3}+\frac{\frac{3}{2} \cdot \frac{4}{2}}{1^3+2^3+3^3}+\ldots$ upto $n$ term is

Given that $a, b$ and $c$ are in AP. The determinant $\left|\begin{array}{lll}x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c\end{array}\right|$ in its simplest form is equal to

The sum of $n$ terms of the series $\frac{4}{1 \cdot 2 \cdot 3}+\frac{5}{2 \cdot 3 \cdot 4}+\frac{6}{3 \cdot 4 \cdot 5}+\ldots$ is

Three arithmetic means between 3 and 19 are

If $a, b$ and $c$ are three positive real numbers, then the mini- mum value of the expression $\frac{b+c}{a}+\frac{c+a}{b}+\frac{a+b}{c}$ is

The sum of the first and fifth terms of an AP is 26 and the product of the second and fourth is 160 . Then the sum of the first six terms of the progression is

Let $S(n)$ denotes the sum of first $n$ terms of an AP. Then, the value of $\frac{S(3 n)}{S(2 n)-S(n)}$ is

If A.M. between $p$ and $q(p \geq q)$ is two times the GM, then $p: q$ is-

If ${a_1},\;{a_2},\;{a_3}, \ldots ,\;{a_{4001}}$ are terms of an AP such that $\frac{1}{{{a_1}{a_2}}} + \frac{1}{{{a_2}{a_3}}} + ... + \frac{1}{{{a_{4000}}{a_{4001}}}} = 10\;$ and ${a_2} + {a_{4000}} = 50,$, then $\left| {{a_1} - {a_{4001}}} \right|$ is equal to

The ratio of sum of $m$ terms to the sum of $n$ terms of an AP is $m^2: n^2$. If $T_k$ is the $k$ th term, then $T_5 / T_2$ is

Let $\mathrm{x}, \mathrm{y}, \mathrm{z}$ be positive real numbers such that $x+y+z=12$ and $x^{3} y^{4} z^{5}=(0.1)(600)^{3}$. Then $x^{3}+y^{3}+z^{3}$ is equal to:

If $a_i \geq 0$ where $i=1,2,3, \ldots \ldots n$ and $\sum\limits_{{i=1}}^n a_i=20$ and if the greatest possible value of $\sum\limits_{{i=1}}^{n-1} a_i\left(a_{i+1}\right)$ is $\mathrm{S}$ then $\mathrm{S}$ is equal to

In the expansion of $\frac{{{e^{7x}} + {e^{3x}}}}{{{e^{5x}}}}$ , the constant term is

The coefficient of ${x^n}$ in the expansion of ${\left( {1 + \frac{{{x^2}}}{{2!}} + \frac{{{x^4}}}{{4!}} + \ldots } \right)^2},$, when $n$ is odd, is

Find the number of positive integral ordered pairs of $(x, y)$ so that $8, x, y$ are in harmonic progression

The arithmetic mean of 7 consecutive integers starting with $a$ is $m$. Then, the arithmetic mean of 11 consecutive integers starting with $a + 2$ is

Let the sequence, ${a_1},{a_2},{a_3}, \ldots ,\;{a_{2n}},$ form an AP, then $a_1^2 - a_2^2 + a_3^2 - ... + a_{2n - 1}^2 - a_{2n}^2$ is equal to

If the A.M. between $a$ and $b$ is $m$ times their H.M. thena : $b=$

The number of numbers lying between 100 and 500 that are divisible by 7 but not by 21 , is

If $1^2+2^2+3^2+\ldots+2003^2=(2003)(4007)(334)$ and $(1)(2003)+(2)(2002)+(3)(2001)$
$+\ldots+(2003)(1)=(2003)(334)(x)$
then $x$ equals

A polygon has 25 sides, the lengths of which starting from the smallest side are in AP. If the perimeter of the polygon is $2100 \mathrm{~cm}$ and the length of the largest side 20 times that of the smallest, then the length of the smallest side and the common difference of an AP are

${e^{x - 1 - \frac{1}{2}{{\left( {x - 1} \right)}^2} + \frac{1}{3}{{\left( {x - 1} \right)}^3} - \frac{1}{4}{{\left( {x - 1} \right)}^4} + \ldots }}$ is equal to

Let the sum of the first $\mathrm{n}$ terms of a non-constant A.P., $\mathrm{a}_1, \mathrm{a}_2, \mathrm{a}_3, \ldots$. be $50 \mathrm{n}+\frac{\mathrm{n}(\mathrm{n}-7)}{2} \mathrm{~A}$, where $\mathrm{A}$ is a constant. If $\mathrm{d}$ is the common difference of this A.P., then the ordered pair $\left(\mathrm{d}, \mathrm{a}_{50}\right)$ is equal to

The value of $\sum\limits_{{r=1}}^n \frac{1}{\sqrt{a+r x}+\sqrt{a+(r-1) x}} \text { is }$

If $\cos x=b$, then for what $b$ do the roots of the equation form an AP?

If the sum of $n$ terms of an $A P$ is $3 n^2+5 n$, then the common difference is

If ${a_n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \ldots + \frac{1}{{{2^n} - 1}},$ then

Let $a^2, b^2, c^2$ be three distinct numbers in A.P. If $a b+b c+c a=1$ then $(b+c),(c+a)$ and $(a+b)$ are in:

If $a,b,c$ are in G.P. and $a - b,c - a$ and $b - c$ are in H.P., then $a + 4b + c$ is equal to