Arithmetic-Geometric Progression (A.G.P.)

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JEE Main 2014

LEVELJEE Main

If $(10)^{9} + 2 (11)^{1} (10)^{8} + 3 (11)^{2} (10)^{7} + \dots + 10 (11)^{9} = k (10)^{9}$ , then $k$ is equal to:

The sum to infinite term of the series $1 + \frac{2}{3} + \frac{6}{3 ^{2}} + \frac{10}{3 ^{3}} + \frac{14}{3 ^{4}} + \dots$ is

The value of $2^{1/4} \cdot 4^{1/8} \cdot 8^{1/16} \dots \infty$ is

If total number of runs scored in $n$ matches is $\frac{n + 1}{4} (2^{n + 1} - n - 2)$ where $n > 1$ , and the runs scored in the $k^{t h}$ match are given by $k \cdot 2^{n + 1 - k}$ , where $1 \leq k \leq n$ . Find $n$ .

JEE Main 2019 (08 April Shift 2)

LEVELJEE Main

The sum $\sum_{k = 1}^{20} \frac{k}{2 ^{k}}$ is equal to-

(A)

2 - \frac{3}{2 ^{17}}

(B)

2 - \frac{11}{2 ^{19}}

(C)

1 - \frac{11}{2 ^{20}}

(D)

2 - \frac{21}{2 ^{20}}

JEE Main 2021 (31 August Shift 2)

LEVELJEE Main

If $S = \frac{7}{5} + \frac{9}{5 ^{2}} + \frac{13}{5 ^{3}} + \frac{19}{5 ^{4}} + \dots$ , then $160 S$ is equal to .

JEE Main 2021 (25 July Shift 1)

LEVELJEE Main

If the value of $(1 + \frac{2}{3} + \frac{6}{3 ^{2}} + \frac{10}{3 ^{3}} + \dots upto \infty)^{l o g_{(0.25)} (\frac{1}{3} + \frac{1}{3 ^{2}} + \frac{1}{3 ^{3}} + \dots upto \infty)}$ is $l$ , then $l^{2}$ is equal to

JEE Main 2021 (26 February Shift 1)

LEVELJEE Main

The sum of the infinite series $1 + \frac{2}{3} + \frac{7}{3 ^{2}} + \frac{12}{3 ^{3}} + \frac{17}{3 ^{4}} + \frac{22}{3 ^{5}} + \dots$ is equal to

(A)

\frac{9}{4}

(B)

\frac{15}{4}

(C)

\frac{13}{4}

(D)

\frac{11}{4}

JEE Main 2022 (25 June Shift 2)

LEVELBoard

The sum $1 + 2 \cdot 3 + 3 \cdot 3^{2} + \dots + 10 \cdot 3^{9}$ is equal to

(A)

\frac{2 \cdot 3 ^{12} + 10}{4}

(B)

\frac{19 \cdot 3 ^{10} + 1}{4}

(C)

5 \cdot 3^{10} - 2

(D)

\frac{9 \cdot 3 ^{10} + 1}{2}

JEE Main 2022 (27 June Shift 2)

LEVELJEE Main

Let $S = 2 + \frac{6}{7} + \frac{12}{7 ^{2}} + \frac{20}{7 ^{3}} + \frac{30}{7 ^{4}} + \dots$ . then $4 S$ is equal to

(A)

(\frac{7}{3})^{2}

(B)

\frac{7 ^{3}}{3 ^{2}}

(C)

(\frac{7}{3})^{3}

(D)

\frac{7 ^{2}}{3 ^{3}}

JEE Main 2022 (29 June Shift 2)

LEVELJEE Main

The sum of the infinite series $1 + \frac{5}{6} + \frac{12}{6 ^{2}} + \frac{22}{6 ^{3}} + \frac{35}{6 ^{4}} + \frac{51}{6 ^{5}} + \frac{70}{6 ^{6}} + \dots$ is equal to:

(A)

\frac{425}{216}

(B)

\frac{429}{216}

(C)

\frac{288}{125}

(D)

\frac{280}{125}

JEE Main 2022 (28 July Shift 1)

LEVELJEE Advanced

Let $x_{1}, x_{2}, x_{3}, \dots, x_{20}$ be in geometric progression with $x_{1} = 3$ and the common ratio $\frac{1}{2}$ . A new data is constructed replacing each $x_{i}$ by $(x_{i} - i)^{2}$ . If $\overset{x}{ˉ}$ is the mean of new data, then the greatest integer less than or equal to $\overset{x}{ˉ}$ is ________.

JEE Main 2022 (29 July Shift 1)

LEVELJEE Main

Let $a_{1}, a_{2}, a_{3}, \dots$ be an AP. If $\sum_{r = 1}^{\infty} \frac{a _{r}}{2 ^{r}} = 4$ , then $4 a_{2}$ is equal to ________.

JEE Main 2023 (06 Apr Shift 2)

LEVELJEE Main

If $(20)^{19} + 2 (21) (20)^{18} + 3 (21)^{2} (20)^{17} + \dots + 20 (21)^{19} = k (20)^{19}$ , then $k$ is equal to ____.

JEE Main 2023 (10 Apr Shift 2)

LEVELJEE Main

Suppose $a_{1}, a_{2}, 2, a_{3}, a_{4}$ be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum of all 5 terms of the arithmetico-geometric progression is $\frac{49}{2}$ , then $a_{4}$ is equal to ______________

JEE Main 2023 (11 Apr Shift 1)

LEVELJEE Main

Let $S = 109 + \frac{108}{5} + \frac{107}{5 ^{2}} + \dots + \frac{2}{5 ^{107}} + \frac{1}{5 ^{108}}$ . Then the value of $(16 S - (25)^{- 54})$ is equal to

JEE Main 2024 (06 Apr Shift 2)

LEVELJEE Main

If $S (x) = (1 + x) + 2 (1 + x)^{2} + 3 (1 + x)^{3} + \dots + 60 (1 + x)^{60}, x \neq = 0$ , and $(60)^{2} S (60) = a (b)^{b} + b$ , where $a, b \in N$ , then $(a + b)$ equal to

JEE Main 2024 (27 Jan Shift 1)

LEVELJEE Main

If $8 = 3 + \frac{1}{4} (3 + p) + \frac{1}{4 ^{2}} (3 + 2 p) + \frac{1}{4 ^{3}} (3 + 3 p) + \dots \infty$ , then the value of $p$ is_____

JEE Main 2025 (January)

LEVELBoard

If $7 = 5 + \frac{1}{7} (5 + α) + \frac{1}{7 ^{2}} (5 + 2 α) + \frac{1}{7 ^{3}} (5 + 3 α) + \dots \infty$ , then the value of $α$ is :

(A)

\frac{6}{7}

(B)

(C)

\frac{1}{7}

(D)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If (10)9+2(11)1(10)8+3(11)2(10)7+⋯+10(11)9=k(10)9, then k is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum to infinite term of the series 1+32​+326​+3310​+3414​+… is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of 21/4⋅41/8⋅81/16…∞ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If total number of runs scored in n matches is 4n+1​(2n+1−n−2) where n>1, and the runs scored in the kth match are given by k⋅2n+1−k, where 1≤k≤n. Find n.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum ∑k=120​2kk​ is equal to-

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If S=57​+529​+5313​+5419​+…, then 160S is equal to .

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the value of (1+32​+326​+3310​+… upto ∞)log(0.25)​(31​+321​+331​+… upto ∞) is l, then l2 is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the infinite series 1+32​+327​+3312​+3417​+3522​+… is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum 1+2⋅3+3⋅32+⋯+10⋅39 is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S=2+76​+7212​+7320​+7430​+…. then 4S is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the infinite series 1+65​+6212​+6322​+6435​+6551​+6670​+… is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a1​,a2​,a3​,… be an AP. If ∑r=1∞​2rar​​=4, then 4a2​ is equal to ________.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If (20)19+2(21)(20)18+3(21)2(20)17+⋯+20(21)19=k(20)19, then k is equal to ____.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S=109+5108​+52107​+⋯+51072​+51081​. Then the value of (16S−(25)−54) is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If S(x)=(1+x)+2(1+x)2+3(1+x)3+⋯+60(1+x)60,x=0, and (60)2S(60)=a(b)b+b, where a,b∈N, then (a+b) equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 8=3+41​(3+p)+421​(3+2p)+431​(3+3p)+…∞, then the value of p is_____

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 7=5+71​(5+α)+721​(5+2α)+731​(5+3α)+…∞, then the value of α is :

If $(10)^{9} + 2 (11)^{1} (10)^{8} + 3 (11)^{2} (10)^{7} + \dots + 10 (11)^{9} = k (10)^{9}$ , then $k$ is equal to:

The sum to infinite term of the series $1 + \frac{2}{3} + \frac{6}{3 ^{2}} + \frac{10}{3 ^{3}} + \frac{14}{3 ^{4}} + \dots$ is

The value of $2^{1/4} \cdot 4^{1/8} \cdot 8^{1/16} \dots \infty$ is

If total number of runs scored in $n$ matches is $\frac{n + 1}{4} (2^{n + 1} - n - 2)$ where $n > 1$ , and the runs scored in the $k^{t h}$ match are given by $k \cdot 2^{n + 1 - k}$ , where $1 \leq k \leq n$ . Find $n$ .

The sum $\sum_{k = 1}^{20} \frac{k}{2 ^{k}}$ is equal to-

If $S = \frac{7}{5} + \frac{9}{5 ^{2}} + \frac{13}{5 ^{3}} + \frac{19}{5 ^{4}} + \dots$ , then $160 S$ is equal to .

If the value of $(1 + \frac{2}{3} + \frac{6}{3 ^{2}} + \frac{10}{3 ^{3}} + \dots upto \infty)^{l o g_{(0.25)} (\frac{1}{3} + \frac{1}{3 ^{2}} + \frac{1}{3 ^{3}} + \dots upto \infty)}$ is $l$ , then $l^{2}$ is equal to

The sum of the infinite series $1 + \frac{2}{3} + \frac{7}{3 ^{2}} + \frac{12}{3 ^{3}} + \frac{17}{3 ^{4}} + \frac{22}{3 ^{5}} + \dots$ is equal to

The sum $1 + 2 \cdot 3 + 3 \cdot 3^{2} + \dots + 10 \cdot 3^{9}$ is equal to

Let $S = 2 + \frac{6}{7} + \frac{12}{7 ^{2}} + \frac{20}{7 ^{3}} + \frac{30}{7 ^{4}} + \dots$ . then $4 S$ is equal to

The sum of the infinite series $1 + \frac{5}{6} + \frac{12}{6 ^{2}} + \frac{22}{6 ^{3}} + \frac{35}{6 ^{4}} + \frac{51}{6 ^{5}} + \frac{70}{6 ^{6}} + \dots$ is equal to:

Let $a_{1}, a_{2}, a_{3}, \dots$ be an AP. If $\sum_{r = 1}^{\infty} \frac{a _{r}}{2 ^{r}} = 4$ , then $4 a_{2}$ is equal to ________.

If $(20)^{19} + 2 (21) (20)^{18} + 3 (21)^{2} (20)^{17} + \dots + 20 (21)^{19} = k (20)^{19}$ , then $k$ is equal to ____.

Let $S = 109 + \frac{108}{5} + \frac{107}{5 ^{2}} + \dots + \frac{2}{5 ^{107}} + \frac{1}{5 ^{108}}$ . Then the value of $(16 S - (25)^{- 54})$ is equal to

If $S (x) = (1 + x) + 2 (1 + x)^{2} + 3 (1 + x)^{3} + \dots + 60 (1 + x)^{60}, x \neq = 0$ , and $(60)^{2} S (60) = a (b)^{b} + b$ , where $a, b \in N$ , then $(a + b)$ equal to

If $8 = 3 + \frac{1}{4} (3 + p) + \frac{1}{4 ^{2}} (3 + 2 p) + \frac{1}{4 ^{3}} (3 + 3 p) + \dots \infty$ , then the value of $p$ is_____

If $7 = 5 + \frac{1}{7} (5 + α) + \frac{1}{7 ^{2}} (5 + 2 α) + \frac{1}{7 ^{3}} (5 + 3 α) + \dots \infty$ , then the value of $α$ is :