Sample QuestionsSequences and Series questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The minimum value of the expression $3^x+3^{1-x}$, $x \in R$, is
- A
$0$
- B
$\frac{1}{3}$
- C
- ✓
$2 \sqrt{3}$
Answer: D.
View full solution →Let $S$ be the sum, $P$ be the product and $R$ be the sum of the reciprocals of 3 terms of a G.P. then $P^2 R^3$ : $S^3$ in equal to
Answer: A.
View full solution →In a G.P. of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the G.P. is
- A
$\frac{-4}{5}$
- B
$\frac{1}{5}$
- ✓
- D
Answer: C.
View full solution →In a G.P. of positive terms, if any term in equal to the sum of the next two terms. Then the common ratio of the G.P. is
Answer: A.
View full solution →If the sum of $n$ terms of an A.P. is given by $S_n=3 n$ $+2 n^2$, then the common difference of the A.P. is
Answer: D.
View full solution →Prove that if $A$ and $G$ be the A.M. and G.M. between two positive number then the numbers are $A \pm \sqrt{A^2-G^2}$
View full solution →Find the number of squares that can be formed on $8 \times 8$ chess board ?
View full solution →Write the $n^{\text {th }}$ term of the series$
\frac{3}{7.11^2}+\frac{5}{8.12^2}+\frac{7}{9.13^2}+\ldots
$
View full solution →If $a, b, c$ are in G.P, then show that $a^2+b^2, a b+b c$, $b^2+c^2$ are also in G.P.
View full solution →Find the sum of the G.P. 0.15,0.015,0.0015,. . . . up to 20 terms.
Find the sum of the series
$1+(1+x)+\left(1+x+x^2\right)+\left(1+x+x^2+x^3\right)+\ldots$
View full solution →If the number $a^2, b^2, c^2$ are given to be in A.P. Show that $\frac{1}{b+c}, \frac{1}{c+a}, \frac{1}{a+b}$ are in A.P.
View full solution →In a G.P. , the $3^{\text {rd }}$ term is 24 and $6^{\text {th }}$ term is 192. Find the 10th term.
View full solution →If in an A.P. $\frac{a_7}{a_{10}}=\frac{5}{7}$, find $\frac{a_4}{a_7}$
View full solution →The first term of an A.P. is $a$ and the sum of the first $p$ terms is zero, show that the sum of its next $q$ terms is $\frac{-a(p+q) q}{p-1}$.
View full solution →Find the sum of the series, $1.3 .4+5.7 .8+9.11 .12+\ldots$ upto $n$ terms.
View full solution →If $p^{\text {th }}, q^{\text {th }}$ and $r^{\text {th }}$ terms of an A.P. and G.P. are both $a, b$ and $c$ respectively, then show that $a^{b-c}$. $b^{c-a} \cdot c^{a-b}=1$.
View full solution →Let $S$ be the sum, $P$ be the product and $R$ be the sum of reciprocals of $n$ terms in a G.P. Prove that $P^2 R^n=S^n$.
View full solution →If $p, q, r$ are in G.P. and the equations $p x^2+2 q x+$ $r=0$ and $d x^2+2 e x+f=0$ have a common root, then show that $\frac{d}{p}, \frac{e}{q}, \frac{f}{r}$ are in A.P.
View full solution →The sum of an infinite G.P. is 57 and the sum of the cubes of its term is 9747 , find the G.P.
The value of $5^{\frac{1}{2}} \cdot 5^{\frac{1}{4}} \cdot 5^{\frac{1}{8}}$. . . . up to infinity is _________________
View full solution →If in a G.P., $a_3+a_5=90$ and if $r=2$, then the first term of G.P. is _________________
View full solution →The third term of a G.P. is 4, the product of first five terms is _________________
If $a, b$ and $c$ are in G.P., then the value of $\frac{a-b}{b-c}$ is equal to _________________
View full solution →If a, b, c are in A.P., then 2b = _________________
If $a$ is A.M. of $b$ and $c$ and the two geometric means are $G_1$ and $G_2$, then prove that $G_1^3+G_2^3=2 a b c $
View full solution →The product of first three terms of G.P. is 1000 . If 6 is added to its second term and 7 is added to its third term, the terms become in A.P. Find G.P.
If $a, b, c$ are in A.P. show that $a\left(\frac{1}{b}+\frac{1}{c}\right), b\left(\frac{1}{c}+\frac{1}{a}\right)$, $c\left(\frac{1}{a}+\frac{1}{b}\right)$ are also in A.P.
View full solution →In an A.P., if the $p$ th term is $\frac{1}{q}$ and $q$ th term is $\frac{1}{p}$.
Prove that the sum of first $p q$ term is $\frac{1}{2}(p q+1)$.
View full solution →The ratio of the sum of $n$ terms of two A.P.'s is $(7 n-1):(3 n+11)$. Find the ratio of their $10^{\text {th }}$ terms.
View full solution →| Column - I | Column - II |
| (a) $1^2+2^2+3^2+\ldots+n^2$ | (i) $\left(\frac{n(n+1)}{2}\right)^2$ |
| (b) $1^3+2^3+3^3+\ldots+n^3$ | (ii) n(n+1) |
| (c) $2+4+6+\ldots+2 n$ | (iii) $\frac{n(n+1)(2 n+1)}{6}$ |
| (d) $1+2+3+\ldots+n$ | (iv) $\frac{n(n+1)}{2}$ |
| Column - I | Column - II |
| (a) 2, 3, 5, 7, . . . | (i) series |
| (b) 13, 8, 3, -2, -7, . . . | (ii) sequence |
| (c) 1 + 2 + 3 + 4 + . . . | (iii) A.P. |
| (d) 1, 3, 5, 7, . . . | (iv) Sequence with $a_n=2 n-1$ |