Question 1: Find the sum of the following arithmetic progressions:

i) to terms

ii) to terms

iii) to terms

iv) to terms

v) to terms

vi) to terms

vii) to terms

Answer:

i) Given series

Given

We know

ii) Given series

Given

We know

iii) Given series

Given

We know

iv) Given series

Given

We know

v) Given series

Given

We know

vi) Given series

Given

We know

vii) Given series

Given

We know

Question 2: Find the sum of the following series:

i)

ii)

iii)

Answer:

i) Given series

Therefore

We know

We know

ii) Given series

Therefore

We know

We know

iii) Given series

Therefore

We know

We know

Question 3: Find the sum of first natural numbers.

Answer:

Given series

Therefore

We know

Question 4: Find the sum of all natural numbers between and , which are divisible by or .

Answer:

Sequence of natural numbers between and , which are divisible by is

Therefore

We know

We know

Sequence of natural numbers between and , which are divisible by is

Therefore

We know

We know

Sequence of natural numbers between and , which are divisible by is

Therefore

We know

We know

Hence, sum of all natural numbers between and , which are divisible by or

Question 5: Find the sum of first odd natural numbers.

Answer:

The series would be

Therefore

We know

Question 6: Find the sum of all odd numbers between and .

Answer:

The series is

Therefore

We know

We know

Question 7: Show that the sum of all odd integers between and which are divisible by is .

Answer:

The series is

Therefore

We know

We know

Question 8: Find the sum of all integers between and , which are multiples of .

Answer:

The series is

Therefore

We know

We know

Question 9: Find the sum of all integers between and which are divisible by .

Answer:

The series is

Therefore

We know

We know

Question 10: Find the sum of all even integers between and .

Answer:

The series is

Therefore

We know

We know

Question 11: Find the sum of all integers between and , which are divisible by .

Answer:

The series is

Therefore

We know

We know

Question 12: Find the sum of the series: to terms.

Answer:

Given sequence

This can we written as

For

For

For

Sum

Question 13: Find the sum of all those integers between and each of which on division leaves the remainder

Answer:

The number would be of the form

The series is

Therefore

We know

We know

Question 14: Solve: (i) (ii) .

Answer:

(i)

Let the number of terms be

We know

Since

We know

(ii)

Let the number of terms be

We know

or

Since

We know

Question 15: Find the term of an A.P., the sum of whose first terms is .

Answer:

Let the first term and the common difference

Given

Question 16: How many terms are there in the A.P. whose first and fifth terms are and respective and the sum of the terms is ?

Answer:

Let the first term and the common difference

Given

We know

or (this is not possible)

Therefore there are terms in the A.P.

Question 17: The sum of first terms of an A.P. is and that of next terms is . Find the progression.

Answer:

Given

… … … … … i)

Also

… … … … … ii)

Solving i) and ii)

Therefore the series is

Question 18: The third term of an A .P. is and the seventh term exceeds three times the third term by . Find the first term, the common difference and the sum of first term.

Answer:

Given

We know

… … … … … i)

Also

… … … … … ii)

Solving i) and ii) we get

We know

and

Question 19: The first term of an A.P. is and the last term is . The sum of all these terms is . Find the common difference.

Answer:

Given

… … … … … i)

We know

Therefore common difference is .

Question 20: The number of terms of an A.P. is even; the sum of odd terms is , of the even terms is , and the last term exceeds the first by , find the number of term and the series.

Answer:

Let the total number of terms be

Let … … … … … i)

… … … … … ii)

Subtracting i) from ii) we get

Also

Therefore there are terms in the series.

Now

Therefore series is

Question 21: If and, , in an A.P., prove that .

Answer:

Given

… … … … … i)

Also

… … … … … ii)

Solving i) and ii) we get

… … … … … iii)

Substituting back in i) we get

Question 22: If term of an A.P. is and the sum of the first four terms is , what is the sum of first terms ?

Answer:

Given and

We know

… … … … … i)

We know

… … … … … ii)

Solving i) and ii) we get

Question 23: It the and terms of an A.P. are and respectively, what is the sum of the first terms ?

Answer:

Given and

We know

… … … … … i)

Similarly,

… … … … … ii)

Solving i) and ii) we get

Question 24: Find the sum of terms of the A.P. whose terms is .

Answer:

Given

For

For

Question 25: Find the sum of all two digit numbers which when divided by , yields as remainder.

Answer:

The number would be of the form

Therefore the series would be for a two digit number

We know

We know

Question 26: If the sum of a certain number of terms of the A.P. is . Find the last term.

Answer:

Given series

We know

Since

Question 27: Find the sum of odd integers from to .

Answer:

The series will be

We know

We know

Question 28: How many terms of the A.P. are needed to give the sum .

Answer:

Given series

We know

or

Question 29: In an A.P. the first term is and the sum of the first five terms is one fourth of the next five terms. Show that term is .

Answer:

Given

We know

… … … … … i)

Similarly, … … … … … ii)

Substituting in i)

We know

Question 30: If be the sum of terms of an A.P. and be the sum of its odd terms, then prove that: .

Answer:

Let the A.P. be

… … … … … i)

Similarly,

… … … … … ii)

. Hence proved.

Question 31: Find an A.P. in which the sum of any number of terms is always three times the squared number of these terms.

Answer:

Given

For

For

For

series is

Question 32: If the sum of n terms of an A.P. is , where and are constants, find the common difference.

Answer:

Given

For

For

Question 33: The sums of terms of two arithmetic progressions are in the ratio . Find the ratio of their terms.

Answer:

Let for first A.P. and

Let for second A.P. and

Given

If you put we get

Question 34: The sums of first terms of two A.P.’s are in the ratio . Find the ratio of their terms.

Answer:

Let for first A.P. and

Let for second A.P. and

Given

If you put we get