Question 1: Find:

i) term of A.P ii) term of A.P

iii) term of A.P

Answer:

i) Given A.P. series

Therefore

Common difference

We know,

ii) Given A.P. series

Therefore

Common difference

We know,

iii) Given A.P. series

Therefore

Common difference

We know,

Question 2: In an A.P., show that .

Answer:

Let the first term be and the common difference be

LHS

. Hence proved.

Question 3:

i) Which term of the A.P. is ?

ii) Which term of the A.P. is ?

iii) Which term of the A.P. is ?

Answer:

i) Given A.P. series is ?

We know

Hence is the term.

ii) Given A.P. series is ?

We know

Hence is the term.

iii) Given A.P. series is ?

We know

Hence is the term.

Question 4:

(i) Is a term of the A.P. ?

(ii) Is a term of the A.P. ?

Answer:

(i) Given A.P. series ?

We know

Since is not a natural number, is NOT a term in the given A.P.

(ii) Given A.P. series ?

We know

Since is not a natural number, is NOT a term in the given A.P.

Question 5:

i) Which term of the sequence is the first negative term?

ii) Which term of the sequence is (a) purely real (b) purely imaginary ?

Answer:

i) Given series

Let the first negative term is .

Therefore

Therefore

Thus the term is the first negative term of the given AP.

ii) Given series

Let the real term be

a) for to be real,

b) for to be imaginary,

Question 6:

i) How many terms are there in the A.P. ?

ii) How many terms are there in the A.P. ?

Answer:

i) Given A.P. series

Therefore

We know

Therefore there are terms in the given A.P.

ii) Given A.P. series ?

Therefore

We know

Therefore there are terms in the given A.P.

Question 7: The first term of an A.P. is , the common difference is and the last term is ; find the number of terms.

Answer:

Given:

We know

Therefore there are terms in the given A.P.

Question 8: The and terms of an A.P. are and respectively, find the term.

Answer:

Given and

We know

… … … … … i)

… … … … … ii)

Solving i) and ii)

Therefore

Hence is

Question 9: If term of an A.P. is zero, prove that its term is double the term.

Answer:

Given:

To prove:

Substituting

. Hence proved.

Question 10: If times the term of an A.P. is equal to times the term, show that term of the A.P. is zero.

Answer:

Given

We know

Now . Hence proved.

Question 11: The and terms of an A.P. are and respectively. Find term.

Answer:

Given and

… … … … … i)

Also

… … … … … ii)

Solving i) and ii) we get

Substituting from i) we get

Therefore

Question 12: In a certain A.P. the term is twice the term. Prove that the term is twice the term.

Answer:

Given

… … … … … i)

To prove:

Substituting from i) we get

. Hence proved.

Question 13: lf term of an A.P. is twice the term, prove that term is twice the term.

Answer:

Given

… … … … … i)

To prove

Substituting from i)

. Hence proved.

Question 14: If the term of the A.P. is same as the term of the A.P. find .

Answer:

term of sequence

term of sequence

Given:

Question 15: Find the term from the end of the following arithmetic progressions: (i) (ii) (iii)

Answer:

(i) Given series

term from the end

Therefore term from the end

(ii) Given series

term from the end

Therefore term from the end

(iii) Given series

term from the end

Therefore term from the end

Question 16: The term of an A.P. is three times the first and the term exceeds twice the third term by . Find the first term and the common difference.

Answer:

Given

… … … … … i)

Also

… … … … … ii)

Substituting in i) we get

Question 17: Find the second term and term of an A.P. whose term is and the term is .

Answer:

We know

Given,

… … … … … i)

Also

… … … … … ii)

Solving i) and ii) we get

Hence

Question 18: How many numbers of two digit are divisible by ?

Answer:

The sequence would be

Therefore

We know

Therefore there are such numbers which are two digit are divisible by

Question 19: An A.P. consists of terms. If the first and the last terms be and respectively, find term.

Answer:

Given

We know

We know

Question 20: The sum of and terms of an A.P. is and the sum of the and terms is . Find the first term and the common difference of the A.P.

Answer:

Given

Also

Solving i) and ii) we get

Substituting in i) we get

Question 21: How many numbers are there between and which when divided by leave remainder ?

Answer:

If a number is divided by and leaves as remainder, it can be represented as

Therefore the sequence would be

Hence

We know

Hence there are numbers between and which when divided by leave remainder

Question 22: The first and the last terms of an A.P. are and respectively. Show that the sum of term from the beginning and term from the end is .

Answer:

Given: first term and last term

term from end

Therefore the sum of the two terms

Question 23: If an A.P. is such that , find .

Answer:

Given

Therefore

Question 24: If are in AP, whose common difference is , show that,

Answer:

Given are in AP

LHS

RHS. Hence proved.