Test Index

Sequence and Series

Question : 30 of 86

Marks: +1, -0

( {P}+1)

( {q}+1)

( {P}+ {q}+1)

(3 {P}+1)

th term is:

Solution:

Concept:

Express terms of an arithmetic progression (AP) using first term

a

and common difference

d

, then use the given condition to find a relation between

a

and

d

Explanation:

Let the AP have first term

a

and common difference

d

The

n

th term is

a + (n-1)d

Given:

(P+1)

th term =

a + Pd

(q+1)

th term =

a + qd

Condition:

a + Pd = 2(a + qd) \implies a + Pd = 2a + 2qd \implies 0 = a + (2q - P)d \implies a = (P - 2q)d

Now,

(P+q+1)

th term =

a + (P+q)d = (P-2q)d + (P+q)d = (2P - q)d

(3P+1)

th term =

a + 3Pd = (P-2q)d + 3Pd = (4P - 2q)d = 2(2P - q)d

Thus the required ratio is

(2P - q)d : 2(2P - q)d = 1:2

(assuming

2P - q \neq 0

Answer:

1:2

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