Arithmetic and geometric progressions — practice questions

Question 1 · difficulty L2 · understanding

Let $\mathrm{A}_1, \mathrm{~A}_2, \mathrm{~A}_3, \ldots \ldots . ., \mathrm{A}_{39}$ be 39 arithmetic means between the numbers 59 and 159. Then the mean of $\mathrm{A}_{25}, \mathrm{~A}_{28}, \mathrm{~A}_{31}$ and $\mathrm{A}_{36}$ is equal to :

• A. 129
• B. 136
• C. 131.50
• D. 134

Question 2 · difficulty L2 · understanding

In an arithmetic progression, if $\mathrm{S}_{40}=1030$ and $\mathrm{S}_{12}=57$, then $\mathrm{S}_{30}-\mathrm{S}_{10}$ is equal to :

• A. 525
• B. 505
• C. 510
• D. 515

Question 3 · difficulty L2 · understanding

Let $$S_n$$ denote the sum of first $$n$$ terms of an arithmetic progression. If $$S_{20}=790$$ and $$S_{10}=145$$, then $$\mathrm{S}_{15}-\mathrm{S}_5$$ is :

• A. 405
• B. 390
• C. 410
• D. 395

Question 4 · difficulty L2 · understanding

$$\text { The } 20^{\text {th }} \text { term from the end of the progression } 20,19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots,-129 \frac{1}{4} \text { is : }$$

• A. $$-115$$
• B. $$-100$$
• C. $$-110$$
• D. $$-118$$

Question 5 · difficulty L2 · understanding

Let S n denote the sum of first n-terms of an arithmetic progression. If S 10 = 530, S 5 = 140, then S 20 $$-$$ S 6 is equal to:

• A. 1862
• B. 1842
• C. 1852
• D. 1872

Question 6 · difficulty L2 · understanding

Let $a_1, a_2, \ldots, a_{2024}$ be an Arithmetic Progression such that $a_1+\left(a_5+a_{10}+a_{15}+\ldots+a_{2020}\right)+a_{2024}=2233$. Then $a_1+a_2+a_3+\ldots+a_{2024}$ is equal to _________.

Question 7 · difficulty L3 · understanding

Let $\alpha=3+4+8+9+13+14+\ldots$ upto 40 terms. If $(\tan \beta)^{\frac{\alpha}{1020}}$ is a root of the equation $x^2+x-2=0, \beta \in\left(0, \frac{\pi}{2}\right)$, then $\sin ^2 \beta+3 \cos ^2 \beta$ is equal to :

• A. ${ }2$
• B. ${\frac{7}{4}}$
• C. $\frac{5}{2}$
• D. $\frac{3}{2}$

Question 8 · difficulty L3 · understanding

Consider the quadratic equation $\left(n^2-2 n+2\right) x^2-3 x+\left(n^2-2 n+2\right)^2=0, n \in \mathbf{R}$. Let $\alpha$ be the minimum value of the product of its roots and $\beta$ be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is $\alpha$ and the common ratio is $\frac{\alpha}{\beta}$, is :

• A. $\frac{61}{37}$
• B. $\frac{121}{81}$
• C. $\frac{364}{243}$
• D. $\frac{1093}{729}$

Question 9 · difficulty L3 · understanding

The sum of the first ten terms of an A.P. is 160 and the sum of the first two terms of a G.P. is 8 . If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:

• A. $\frac{34}{9}$
• B. $\frac{34}{13}$
• C. $\frac{32}{9}$
• D. $\frac{32}{13}$

Question 10 · difficulty L3 · understanding

Let $\alpha, \beta$ be the roots of the equation $x^2-x+\mathrm{p}=0$ and $\gamma, \delta$ be the roots the equation $x^2-4 x+\mathrm{q}=0$; $p, q \in \mathbf{Z}$. If $\alpha, \beta, \gamma, \delta$ are in G.P., then $|p+q|$ equals :

• A. 16
• B. 32
• C. 34
• D. 38