JEE Advanced · Mathematics · 5. Sequences & Series

Let $A P (a; d)$ denote the set of all the terms of an infinite arithmetic progression with first term a and common difference $d > 0 .$ If $A P (1; 3) \cap A P (2; 5) \cap A P (3; 7) = A P (a; d)$ then $a + d$ equals ____

A 150
B 157
C 160
D 187

Verified Solution

Answer & Solution

Correct Answer

(B) 157

Step-by-step Solution

Detailed explanation

Let

T_{(1, m)} :

m^{t h}

term of

1^{s t}

series

A (1; 3)

T_{(2, n)} :

n^{t h}

term of

2^{n d}

series

A (2; 5)

T_{(3, r)} :

r^{t h}

term of

3^{r d}

series

A (3; 7)

For common terms

T_{(1, m)} = T_{(2, n)} = T_{(3, r)}

1 + 3 (m - 1) = 2 + 5 (n - 1) = 3 + 7 (r - 1)

3 m - 2 = 5 n - 3 = 7 r - 4

\Rightarrow m = \frac{5 n - 1}{3} a n d r = \frac{5 n + 1}{7}

For

‘ m ’

to be a natural number

n = 2,5, 11, \dots

And for

‘ r ’

to be a natural number

n = 4,11, \dots

When

n = 11, m = 18

and

r = 8

\Rightarrow

For

1^{s t}

common term of three series

A (a, d)

a = 1^{s t}

term of

A (1,3)

a = 1 + (181) 3 = 52

\Rightarrow

Common difference of

A (a, d) = L C M

of common difference of

(A (1; 3), A (2; 5), A (3; 7))

d = L C M (3, 5, 7)

d = 105

Hence,

a + d = 52 + 105 = 157

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

If $\lim _{x \rightarrow \infty}\left(\frac{x^{2}+x+1}{x+1}-a x-b\right)=4$, then

JEE Advanced 2012 Hard

Consider the ellipse

\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1

. Let

H (α, 0), 0 < α < 2

, be a point. A straight line drawn through

H

parallel to the

y

-axis crosses the ellipse and its auxiliary circle at points

E

and

F

respectively, in the first quadrant. The tangent to the ellipse at the point

E

intersects the positive

x

-axis at a point

G

. Suppose the straight line joining

F

and the origin makes an angle

ϕ

with the positive

x

-axis.

	List- $I$		List- $II$
$(I)$	If $ϕ = \frac{π}{4}$ , then the area of the triangle $F G H$ is	$(P)$	$\frac{{(\sqrt{3} - 1)}^{4}}{8}$
$(II)$	If $ϕ = \frac{π}{3}$ , then the area of the triangle $F G H$ is	$(Q)$	$1$
$(III)$	If $ϕ = \frac{π}{6}$ , then the area of the triangle $F G H$ is	$(R)$	$\frac{3}{4}$
$(IV)$	If $ϕ = \frac{π}{12}$ , then the area of the triangle $F G H$ is	$(S)$	$\frac{1}{2 \sqrt{3}}$
		$(T)$	$\frac{3 \sqrt{3}}{2}$

The correct option is:

JEE Advanced 2022 Easy

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Let AP a;d denote the set of all the terms of an infinite arithmetic progression with first term a and common difference d>0. If AP1;3∩AP2;5∩AP3;7=APa;d then a+d equals ____

Step-by-step Solution

Let $A P (a; d)$ denote the set of all the terms of an infinite arithmetic progression with first term a and common difference $d > 0 .$ If $A P (1; 3) \cap A P (2; 5) \cap A P (3; 7) = A P (a; d)$ then $a + d$ equals ____