JEE Main Sequences and Series Practice Questions With Solutions

Let $a,ar,a{r}^2$a , a r , a r^{2}, ............ be an infinite G.P. If $\sum _{n=0}^{\mathrm{\infty }}a{r}^n=57$\sum_{n = 0}^{\infty} a r^{n} = 57 and $\sum _{n=0}^{\mathrm{\infty }}{a}^3{r}^{3n}=9747$\sum_{n = 0}^{\infty} a^{3} r^{3 n} = 9747, then $a+18r$a + 18 r is equal to

If the sum of the series $\frac{1}{1\cdot (1+\mathrm{d})}+\frac{1}{(1+\mathrm{d})(1+2\mathrm{d})}+\dots +\frac{1}{(1+9\mathrm{d})(1+10\mathrm{d})}$\frac{1}{1 \cdot \left(\right. 1 + d \left.\right)} + \frac{1}{\left(\right. 1 + d \left.\right) \left(\right. 1 + 2 \textrm{ } d \left.\right)} + \ldots + \frac{1}{\left(\right. 1 + 9 \textrm{ } d \left.\right) \left(\right. 1 + 10 \textrm{ } d \left.\right)} is equal to 5, then $50\mathrm{d}$50 \textrm{ } d is equal to :

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\frac{70}{3}$\frac{70}{3} and the product of the third and fifth terms is 49. Then the sum of the $4^{\text{th }},6^{\text{th }}$4^{\text{th}\textrm{ }} , 6^{\text{th}\textrm{ }} and $8^{\text{th }}$8^{\text{th}\textrm{ }} terms is equal to:

Let $ABC$A B C be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $ABC$A B C and the same process is repeated infinitely many times. If $\mathrm{P}$P is the sum of perimeters and $Q$Q is be the sum of areas of all the triangles formed in this process, then :

A software company sets up m number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $\mathrm{m}$m is equal to:

For $x⩾0$x \geq 0, the least value of $\mathrm{K}$K, for which $4^{1+x}+4^{1-x},\frac{\mathrm{K}}{2},{16}^x+{16}^{-x}$4^{1 + x} + 4^{1 - x} , \frac{K}{2} , 16^{x} + 16^{- x} are three consecutive terms of an A.P., is equal to :

If $\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\dots +\frac{1}{\sqrt{99}+\sqrt{100}}=m$\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \ldots + \frac{1}{\sqrt{99} + \sqrt{100}} = m and $\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\dots +\frac{1}{99\cdot 100}=\mathrm{n}$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \ldots + \frac{1}{99 \cdot 100} = n, then the point $(\mathrm{m},\mathrm{n})$\left(\right. m , n \left.\right) lies on the line

The value of $\frac{1\times 2^2+2\times 3^2+\dots +100\times (101{)}^2}{1^2\times 2+2^2\times 3+\dots .+{100}^2\times 101}$\frac{1 \times 2^{2} + 2 \times 3^{2} + \ldots + 100 \times \left(\right. 101 \left.\right)^{2}}{1^{2} \times 2 + 2^{2} \times 3 + \ldots . + 100^{2} \times 101} is

Let three real numbers $a,b,c$a , b , c be in arithmetic progression and $a+1,b,c+3$a + 1 , b , c + 3 be in geometric progression. If $a>10$a > 10 and the arithmetic mean of $a,b$a , b and $c$c is 8, then the cube of the geometric mean of $a,b$a , b and $c$c is

Let the first three terms 2, p and q, with $q\ne 2$q \neq 2, of a G.P. be respectively the $7^{\text{th }},8^{\text{th }}$7^{\text{th}\textrm{ }} , 8^{\text{th}\textrm{ }} and ${13}^{\text{th }}$13^{\text{th}\textrm{ }} terms of an A.P. If the $5^{\text{th }}$5^{\text{th}\textrm{ }} term of the G.P. is the $n^{\text{th }}$n^{\text{th}\textrm{ }} term of the A.P., then $n$n is equal to: