Sequence of series practice · Mixed difficulty

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then, the common ratio of this progression equals

The product of $x^{1 / 2} \cdot x^{1 / 4} \cdot x^{1 / 8}$.. Equals

If three positive real numbers $a,\;b,\;c$ are in AP and $abc = 4$, then the minimum possible value of $b$ is

Consider an infinite geometric series with first term $a$ and common ratio $r$. If its sum is 4 and the second term is $3 / 4$, then

If $a_1, a_2 \ldots, a_n$ are positive real numbers whose product is a fixed number $\mathrm{c}$, then the minimum value of $a_1+a_2+\ldots \ldots+a_{n-1}+2 a_n$ is

If S is the sum to infinity of a G.P., whose first term is $a,$ then the sum of the first $n$ terms is

For a sequence $< {a_n} > ,{a_1} = 2$ and $\frac{{{a_{n + 1}}}}{{{a_n}}} = \frac{1}{3}.$ Then, $\sum \limits_{r = 1}^{20} {a_r}$ is

If $\cos (x-y), \cos x$ and $\cos (x+y)$ are in HP, then $\cos x \sec \left(\frac{y}{2}\right)$ is equal to

Every term of a G.P. is positive and also every term is the sum of two preceding terms. Then, the common ratio of the G.P. is

$\sin ^{-1}\left(a-\frac{a^2}{3}+\frac{a^3}{9}+\ldots.\right)+\cos ^{-1}\left(1+b+b^2+\ldots\right)=\frac{\pi}{2}$ when

If ${3^{2x + 1}}{.4^{x - 1}} = 36,$ then $x =$

The GM of roots of the equation ${x^2} - 18x + 9 = 0$ is

Let $x \in \left( {1,\infty } \right)$ and $n$ be a positive integer greater than 1. If ${f_n}\left( x \right) = \frac{n}{{\frac{1}{{{{\log }_2}x}} + \frac{1}{{{{\log }_3}x}} + \ldots + \frac{1}{{{{\log }_n}x}}}}$, then ${\left( {n!} \right)^{{f_n}\left( x \right)}}$ equals to

The sum of the series $1 + 3x + 6{x^2} + 10{x^3} + ...\infty$ will be

If $\log _{10}\left(\frac{1}{2^x+x-1}\right)=x\left(\log _{10} 5-1\right)$ then $x$ is equal to

The sum of $n$ terms of the following series $1 + \left( {1 + x} \right) + \left( {1 + x + {x^2}} \right) + \ldots$ is

If $\frac{{a + b}}{{1 - ab}},b,\frac{{b + c}}{{1 - bc}}$ are in A.P., then $a,\frac{1}{b},c$ are in

If $\sqrt{3}, A$ and $\sqrt{2}$ are in $A P$, then $\frac{\sqrt{3}+\sqrt{2}}{2}$ is greater than or equal to

The sum of series $\frac{1}{{1.2.3}} + \frac{1}{{3.4.5}} + \frac{1}{{5.6.7}} + \ldots \infty$ is equal to

The sum of all two digit numbers which, when divided by 4, yield unity as a remainder is

The sum of the series $1 + \frac{1}{3}.\frac{1}{4} + \frac{1}{5}.\frac{1}{{{4^2}}} + \frac{1}{7}.\frac{1}{{{4^3}}} + \ldots \infty$ is

If $a,b,c$ are in H.P., then $\frac{a}{{b + c}},\frac{b}{{c + a}},\frac{c}{{a + b}}$ will be in

If the sum to $2n$ terms of the AP 2, 5, 8, 11,… is equal to the sum to n terms of the AP 57, 59, 61, 63,…, then $n$ is equal to

If $a, b, c$ are in H.P., then $\frac{a}{b+c-a}, \frac{b}{c+a-b}, \frac{c}{a+b-c}$ are in

If $x = 1 + a + {a^2} + ...\infty$ and $y = 1 + b + {b^2} + ...\infty$ where $a$ and $b$ are proper fractions, then $1 + ab + {a^2}{b^2} + ...\infty$ equals

The sum of all two digit odd numbers is

The eighth term of a geometric progression is 128 and common ratio is 2. The product of the first five terms is

$\sum\limits_{{i=1}}^n \sum\limits_{{j=1}}^i \sum\limits_{{k=1}}^j 1=$

If the sum to first $n$ terms of the AP 2, 4, 6,… is 240, then the value of $n$ is

A club consists of members whose ages are in AP, the common difference being 3 months. If the youngest member of the club is just 7 yr old and the sum of the ages of all the members is $250 \mathrm{yr}$, then the number of members in the club is