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.
If $t_n$ denotes the $n^{th}$ term of the series $2 + 3 + 6 + 11 + 18 + ...$ then $t_{50}$ is:
- A
$49^2 - 1$
- B
$49^2$
- C
$50^2 + 1$
- ✓
$49^2 + 2$
Answer: D.
View full solution →Let $S_n$ denote the sum of the cubes of the first n natural numbers and $s_n$ denote the sum of the first $n$ natural numbers. Then $\sum\limits^\text{n}_{\text{r}=1}\frac{\text{S}_\text{r}}{\text{S}_\text{r}}$ equals to:
- ✓
$\frac{\text{n}(\text{n}+1)(\text{n}+2)}{6}$
- B
$\frac{\text{n}(\text{n}+1)}{2}$
- C
$\frac{\text{n}^{2}+3\text{n}+2}{2}$
- D
Answer: A.
View full solution →Let $S_n$ denote the sum of the first $n$ terms of an $\text{A.P.}$ If $S_{2n} = 3S_n,$ then $S_{3n} : S_n$ is equal to:
Answer: B.
View full solution →The minimum value of $4^{\text{x}}+4^{1-\text{x}},\text{x}\in\text{R}$ is:
Answer: B.
View full solution →If $x, 2y$ and $3z$ are in $\text{A.P.}$ where the distinct numbers $x, y$ and $z$ are in $\text{G.P.,}$ then the common ratio of the $\text{G.P.}$ is:
- A
$3$
- B
$\frac{1}{3}$
- C
$2$
- ✓
$\frac{1}{2}$
Answer: D.
View full solution →If the sum of n terms of a sequence is quadratic expression, then it always represents an $A.P.$
View full solution →Any term of an $A.P. ($except first$)$ is equal to half the sum of terms which are equidistant from it.
View full solution →The sum or difference of two G.P.s, is again a G.P.
Two sequences cannot be in both $A.P.$ and $G.P.$ together.
View full solution →Every progression is a sequence but the converse i.e., every sequence is also a progression need not necessarily be true.
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 $q$ term is $\frac{-\text{a}(\text{p + q})\text{q}}{\text{p}-1}.$
$[$Hint: Required sum $= S_{p + q}- S_p]$
View full solution →The sum of interior angles of a triangle is $180^\circ$ . Show that the sum of the interior angles of polygons with $3, 4, 5, 6,…$ sides form an arithmetic progression. Find the sum of the interior angles for a $21$ sided polygon.
View full solution →Match the questions given under Column I with their appropriate answers given under the Column $II.$
|
|
$\text{Column I} $
|
|
$\text{Column II} $
|
|
$(a)$
|
$4,1,\frac{1}{4},\frac{1}{16}$ |
$(i)$ |
$A.P.$ |
| $(b)$ |
$2, 3, 5, 7$ |
$(ii)$
|
$Squence$
|
| $(c)$ |
$13, 8, 3, -2, -7$ |
$(iii)$ |
$G.P.$ |
View full solution →A man accepts a position with an initial salary of $Rs. 5200$ per month. It is understood that he will receive an automatic increase of $Rs. 320$ in the very next month and each month thereafter.
- Find his salary for the tenth month.
- What is his total earnings during the first year?
View full solution →A man saved Rs. 66000 in 20 years. In each succeeding year after the first year he saved Rs. 200 more than what he saved in the previous year. How much did he save in the first year?
Find the $r^{th}$ term of an $A.P.$ sum of whose first n terms is $2n + 3n^2 $.
$[$Hint: $an = S_n– S_n– 1]$
View full solution →If $a_1, a_2, a_3, ..., an$ are in$ A.P.,$ where $a_i > 0$ for all $i,$ show that:
$\frac{1}{\sqrt{\text{a}_1}+\sqrt{\text{a}_2}}+\frac{1}{\sqrt{\text{a}_2}+\sqrt{\text{a}_3}}+....+\frac{1}{\sqrt{\text{a}_{\text{n}-1}}+\sqrt{\text{a}_\text{n}}}=\frac{\text{n}-1}{\sqrt{\text{a}_1}+\sqrt{\text{a}_\text{n}}}$
View full solution →The third term of a G.P. is 4. The product of the first five terms is _____.
For a, b, c to be in G.P. the value of $\frac{\text{a}-\text{b}}{\text{b}-\text{c}}$ is equal to _____.
View full solution →The sum of terms equidistant from the beginning and end in an $A.P.$ is equal to _____.
View full solution →Match the questions given under Column $I$ with their appropriate answers given under the Column $II.$
| |
Column I |
|
Column II |
| $(a)$ |
$1^2+2^2+3^2+....+\text{n}^2$ |
$(i)$ |
$\Big[\frac{\text{n}(\text{n}+1)}{2}\Big]^2$ |
| $(b)$ |
$1^3+2^3+3^3+....\text{n}^3$ |
$(ii)$ |
$\text{n}(\text{n}+1)$ |
| $(c)$ |
$2+4+6+....+2\text{n}$ |
$(iii)$ |
$\frac{\text{n}(\text{n}+1)(2\text{n}+1)}{6}$ |
| $(d)$ |
$1+2+3+....\text{n}$ |
$(iv)$ |
$\frac{\text{n}(\text{n}+1)}{2}$ |
View full solution →If $\theta_1,\theta_2,\theta_3,....\theta_\text{n}$ are in A.P., whose common difference is d, show that $\sec\theta_1\cdot\sec\theta_2+\sec\theta_2+\sec\theta_3+\dots+\sec\theta_{\text{n}-1}\cdot\sec\theta_\text{n}=\frac{\tan\theta_\text{n}-\tan\theta_1}{\sin\text{d}}$
View full solution →A carpenter was hired to build $192$ window frames. The first day he made five frames and each day, thereafter he made two more frames than he made the day before. How many days did it take him to finish the job?
View full solution →$1$f the sum of p terms of an $AP$. is $q$ and the sum of $q$ terms is $p$, then show that the sum of $p +q$ terms is $-(p + q)$. Also$,$ find the sum of first $p - q$ terms $($where$, p > q).$
View full solution →In a potato race 20 potatoes are placed in a line at intervals of 4 metres with the first potato 24 metres from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?