Arithmetic Progression (A.P.)

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JEE Main 2011

LEVELJEE Main

A man saves Rs $200$ in each of the first three months of his service. In each of the subsequent months his saving increases by Rs $40$ more than the saving of immediately previous month. His total saving from the start of service will be Rs $11040$ after

A person is to count $4500$ currency notes. Let $a_{n}$ denote the number of notes he counts in the $n$ th minute. If $a_{1} = a_{2} = \dots = a_{10} = 150$ and $a_{10}, a_{11}, \dots$ are in an AP with common difference $- 2$ , then the time taken by him to count all notes is

Let $a_{1}, a_{2}, a_{3} \dots$ be terms on A.P. If $\frac{a _{1} + a _{2} + \dots + a _{p}}{a _{1} + a _{2} + \dots + a _{q}} = \frac{p ^{2}}{q ^{2}}, p \neq = q$ , then $\frac{a _{6}}{a _{21}}$ equals

Let $T_{r}$ be the $r$ th term of an A.P. whose first term is $a$ and common difference is $d$ . If for some positive integers $m, n, m \neq = n, T_{m} = 1/ n$ and $T_{n} = 1/ m$ , then $a - d$ equals

(A)

\frac{1}{m} + \frac{1}{n}

(B)

1

(C)

\frac{1}{mn}

(D)

0

JEE Main 2003

LEVELJEE Main

Let $f (x)$ be a polynomial function of second degree. If $f (1) = f (- 1)$ and $a, b, c$ are in A.P, then $f^{'} (a), f^{'} (b), f^{'} (c)$ are in

(A)

Arithmetic-Geometric Progression

If $1, lo g_{9} (3^{1 - x} + 2), lo g_{3} (4 \cdot 3^{x} - 1)$ are in A.P. then $x$ equals

(A)

lo g_{3} 4

(B)

1 - lo g_{3} 4

(C)

1 - lo g_{4} 3

(D)

lo g_{4} 3

JEE Main 2002

LEVELJEE Main

The sum of integers from 1 to 100 that are divisible by 2 or 5 is

Suppose $a, b, c$ are in A.P. and $a^{2}, b^{2}, c^{2}$ are in G.P. if $a < b < c$ and $a + b + c = 3/2$ , then the value of $a$ is

(A)

\frac{1}{2 2}

(B)

\frac{1}{2 3}

(C)

\frac{1}{2} - \frac{1}{3}

(D)

\frac{1}{2} - \frac{1}{2}

JEE Advanced 2001

LEVELBoard

If the sum of the first $2 n$ terms of the A.P. $2, 5, 8, \dots$ is equal to the sum of the first $n$ terms of the A.P. $57, 59, 61, \dots$ , then $n$ equals

Let $T_{r}$ be the $r$ th term of an A.P., for $r = 1, 2, 3, \dots$ . If for some positive integers $m, n$ we have $T_{m} = 1/ n$ and $T_{n} = 1/ m$ , then $T_{mn}$ equals

(A)

\frac{1}{mn}

(B)

\frac{1}{m} + \frac{1}{n}

(C)

1

(D)

0

JEE Advanced 2015

LEVELJEE Main

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is $6 : 11$ and the seventh term lies in between $130$ and $140$ , then the common difference of this A.P. is \dots.

JEE Advanced 2011

LEVELJEE Main

Let $a_{1}, a_{2}, a_{3}, \dots, a_{100}$ be an arithmetic progression with $a_{1} = 3$ and $S_{p} = \sum_{i = 1}^{p} a_{i}, 1 \leq p \leq 100$ . For any integer $n$ with $1 \leq n \leq 20$ , let $m = 5 n$ . If $S_{m} / S_{n}$ does not depend on $n$ , then $a_{2}$ is \dots.

JEE Advanced 2010

LEVELJEE Main

Let $a_{1}, a_{2}, a_{3}, \dots, a_{11}$ be real numbers satisfying $a_{1} = 15, 27 - 2 a_{2} > 0$ and $a_{k} = 2 a_{k - 1} - a_{k - 2}$ for $k = 3, 4, \dots, 11$ . If $\frac{a _{1}^{2} + a _{2}^{2} + \dots + a _{11}^{2}}{11} = 90$ , then the value of $\frac{a _{1} + a _{2} + \dots + a _{11}}{11}$ is equal to \dots.

JEE Advanced 1990

LEVELJEE Main

If $lo g_{3} 2, lo g_{3} (2^{x} - 5)$ , and $lo g_{3} (2^{x} - \frac{7}{2})$ are in arithmetic progression, determine the value of $x$ .

JEE Advanced 1984

LEVELJEE Main

The sum of integers from 1 to 100 that are divisible by 2 or 5 is \dots.

JEE Advanced 1980

LEVELJEE Main

The interior angles of a polygon are in arithmetic progression. The smallest angle is $12 0^{\circ}$ , and the common difference is $5^{\circ}$ , Find the number of sides of the polygon.

JEE Advanced 2000

LEVELJEE Main

The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer.

JEE Advanced 2007

LEVELJEE Main

Comprehension Passage

Let

V_{r}

denote the sum of first

r

terms of an arithmetic progression (A.P.) whose first term is

r

and the common difference is

(2 r - 1)

. Let

T_{r} = V_{r + 1} - V_{r} - 2

and

Q_{r} = T_{r + 1} - T_{r}

for

r = 1, 2, \dots

Question 1:

The sum $V_{1} + V_{2} + \dots + V_{n}$ is

(A)

\frac{1}{12} n (n + 1) (3 n^{2} - n + 1)

(B)

\frac{1}{12} n (n + 1) (3 n^{2} + n + 2)

(C)

\frac{1}{3} n (2 n^{2} - n + 1)

(D)

\frac{1}{3} (2 n^{3} - 2 n + 3)

Question 2:

$T_{r}$ is always

Which one of the following is a correct statement ?

(A)

Q_{1}, Q_{2}, Q_{3}, \dots

are in A.P. with common difference 5

(B)

Q_{1}, Q_{2}, Q_{3}, \dots

are in A.P. with common difference 6

(C)

Q_{1}, Q_{2}, Q_{3}, \dots

are in A.P. with common difference 11

(D)

Q_{1} = Q_{2} = Q_{3} = \dots

JEE Main 2018 (Paper 1)

LEVELJEE Main

Let $a_{1}, a_{2}, a_{3}, \dots, a_{49}$ be in A.P. such that $\sum_{k = 0}^{12} a_{4 k + 1} = 416$ and $a_{9} + a_{43} = 66$ . If $a_{1}^{2} + a_{2}^{2} + \dots + a_{17}^{2} = 140 m$ , then $m$ is equal to :

JEE Main 2019 (9 January)

LEVELJEE Main

Let $a_{1}, a_{2}, ........., a_{30}$ be an A. P., $S = \sum_{i = 1}^{30} a_{i}$ and $T = \sum_{i = 1}^{15} a_{(2 i - 1)}$ . If $a_{5} = 27$ and $S - 2 T = 75$ , then $a_{10}$ is equal to :

JEE Main 2019 (10 January Shift 1)

LEVELBoard

The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is :

JEE Main 2019 (8 April Shift 1)

LEVELJEE Main

The sum of all natural numbers 'n' such that $100 < n < 200$ and H.C.F. $(91, n) > 1$ is :

(A)

3221

(B)

3121

(C)

3203

(D)

3303

JEE Main 2019 (09 April Shift 1)

LEVELBoard

Let the sum of the first $n$ terms of a non-constant A.P., $a_{1}, a_{2}, a_{3}, \dots$ be $50 n + \frac{n ( n - 7 )}{2} A$ , where $A$ is a constant. If $d$ is the common difference of this A.P., then the ordered pair $(d, a_{50})$ is equal to

(A)

(A, 50 + 46 A)

(B)

(A, 50 + 45 A)

(C)

(50, 50 + 46 A)

(D)

(50, 50 + 45 A)

JEE Main 2019 (10 April Shift 1)

LEVELBoard

If $a_{1}, a_{2}, a_{3}, \dots, a_{n}$ are in A.P. and $a_{1} + a_{4} + a_{7} + \dots + a_{16} = 114$ , then $a_{1} + a_{6} + a_{11} + a_{16}$ is equal to :

JEE Main 2019 (12 April Shift 1)

LEVELBoard

Let $S_{n}$ denote the sum of the first n terms of an A.P. If $S_{4} = 16$ and $S_{6} = - 48$ , then $S_{10}$ is equal to :

(A)

- 320

(B)

- 260

(C)

- 380

(D)

- 410

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a1​,a2​,a3​… be terms on A.P. If a1​+a2​+⋯+aq​a1​+a2​+⋯+ap​​=q2p2​,p=q, then a21​a6​​ equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let Tr​ be the rth term of an A.P. whose first term is a and common difference is d. If for some positive integers m,n,m=n,Tm​=1/n and Tn​=1/m, then a−d equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x) be a polynomial function of second degree. If f(1)=f(−1) and a,b,c are in A.P, then f′(a),f′(b),f′(c) are in

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 1,log9​(31−x+2),log3​(4⋅3x−1) are in A.P. then x equals

The sum of integers from 1 to 100 that are divisible by 2 or 5 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Suppose a,b,c are in A.P. and a2,b2,c2 are in G.P. if a<b<c and a+b+c=3/2, then the value of a is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the sum of the first 2n terms of the A.P. 2,5,8,… is equal to the sum of the first n terms of the A.P. 57,59,61,…, then n equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let Tr​ be the rth term of an A.P., for r=1,2,3,…. If for some positive integers m,n we have Tm​=1/n and Tn​=1/m, then Tmn​ equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If log3​2,log3​(2x−5), and log3​(2x−27​) are in arithmetic progression, determine the value of x.

The sum of integers from 1 to 100 that are divisible by 2 or 5 is \dots.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The interior angles of a polygon are in arithmetic progression. The smallest angle is 120∘, and the common difference is 5∘, Find the number of sides of the polygon.

The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer.

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum V1​+V2​+⋯+Vn​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Tr​ is always

Which one of the following is a correct statement ?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a1​,a2​,a3​,…,a49​ be in A.P. such that ∑k=012​a4k+1​=416 and a9​+a43​=66. If a12​+a22​+⋯+a172​=140m, then m is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a1​,a2​,.........,a30​ be an A. P., S=∑i=130​ai​ and T=∑i=115​a(2i−1)​. If a5​=27 and S−2T=75, then a10​ is equal to :

The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of all natural numbers 'n' such that 100<n<200 and H.C.F. (91,n)>1 is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a1​,a2​,a3​,…,an​ are in A.P. and a1​+a4​+a7​+⋯+a16​=114, then a1​+a6​+a11​+a16​ is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let Sn​ denote the sum of the first n terms of an A.P. If S4​=16 and S6​=−48, then S10​ is equal to :

Let $a_{1}, a_{2}, a_{3} \dots$ be terms on A.P. If $\frac{a _{1} + a _{2} + \dots + a _{p}}{a _{1} + a _{2} + \dots + a _{q}} = \frac{p ^{2}}{q ^{2}}, p \neq = q$ , then $\frac{a _{6}}{a _{21}}$ equals

Let $T_{r}$ be the $r$ th term of an A.P. whose first term is $a$ and common difference is $d$ . If for some positive integers $m, n, m \neq = n, T_{m} = 1/ n$ and $T_{n} = 1/ m$ , then $a - d$ equals

Let $f (x)$ be a polynomial function of second degree. If $f (1) = f (- 1)$ and $a, b, c$ are in A.P, then $f^{'} (a), f^{'} (b), f^{'} (c)$ are in

If $1, lo g_{9} (3^{1 - x} + 2), lo g_{3} (4 \cdot 3^{x} - 1)$ are in A.P. then $x$ equals

Suppose $a, b, c$ are in A.P. and $a^{2}, b^{2}, c^{2}$ are in G.P. if $a < b < c$ and $a + b + c = 3/2$ , then the value of $a$ is

If the sum of the first $2 n$ terms of the A.P. $2, 5, 8, \dots$ is equal to the sum of the first $n$ terms of the A.P. $57, 59, 61, \dots$ , then $n$ equals

Let $T_{r}$ be the $r$ th term of an A.P., for $r = 1, 2, 3, \dots$ . If for some positive integers $m, n$ we have $T_{m} = 1/ n$ and $T_{n} = 1/ m$ , then $T_{mn}$ equals

If $lo g_{3} 2, lo g_{3} (2^{x} - 5)$ , and $lo g_{3} (2^{x} - \frac{7}{2})$ are in arithmetic progression, determine the value of $x$ .

The interior angles of a polygon are in arithmetic progression. The smallest angle is $12 0^{\circ}$ , and the common difference is $5^{\circ}$ , Find the number of sides of the polygon.

The sum $V_{1} + V_{2} + \dots + V_{n}$ is

$T_{r}$ is always

Let $a_{1}, a_{2}, a_{3}, \dots, a_{49}$ be in A.P. such that $\sum_{k = 0}^{12} a_{4 k + 1} = 416$ and $a_{9} + a_{43} = 66$ . If $a_{1}^{2} + a_{2}^{2} + \dots + a_{17}^{2} = 140 m$ , then $m$ is equal to :

Let $a_{1}, a_{2}, ........., a_{30}$ be an A. P., $S = \sum_{i = 1}^{30} a_{i}$ and $T = \sum_{i = 1}^{15} a_{(2 i - 1)}$ . If $a_{5} = 27$ and $S - 2 T = 75$ , then $a_{10}$ is equal to :

The sum of all natural numbers 'n' such that $100 < n < 200$ and H.C.F. $(91, n) > 1$ is :

If $a_{1}, a_{2}, a_{3}, \dots, a_{n}$ are in A.P. and $a_{1} + a_{4} + a_{7} + \dots + a_{16} = 114$ , then $a_{1} + a_{6} + a_{11} + a_{16}$ is equal to :

Let $S_{n}$ denote the sum of the first n terms of an A.P. If $S_{4} = 16$ and $S_{6} = - 48$ , then $S_{10}$ is equal to :