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

Answer:

Question 2: Find the sum of the following series:

Answer:

Question 3: Find the sum of first natural numbers.

Answer:

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

Answer:

Sequence of natural numbers between , which are divisible by is

Sequence of natural numbers between , which are divisible by is

Sequence of natural numbers between , which are divisible by is

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

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

Answer:

The series would be

Question 6: Find the sum of all odd numbers between

Answer:

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

Answer:

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

Answer:

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

Answer:

Question 10: Find the sum of all even integers between

Answer:

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

Answer:

Question 12: Find the sum of the series:

Answer:

Given sequence

This can we written as

Sum

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

Answer:

The number would be of the form

Question 14: Solve:

Answer:

Let the number of terms be

Let the number of terms be

or

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

Answer:

Let the first term and the common difference

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

Answer:

Let the first term and the common difference

or (this is not possible)

Therefore there are in the A.P.

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

Answer:

… … … … … i)

… … … … … ii)

Solving i) and ii)

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:

… … … … … i)

… … … … … ii)

Solving i) and ii) we get

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:

… … … … … i)

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

Therefore there are in the series.

Now

Therefore series is

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

Answer:

… … … … … i)

… … … … … 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 ?

Answer:

… … … … … i)

… … … … … ii)

Solving i) and ii) we get

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

Answer:

… … … … … i)

Similarly,

… … … … … ii)

Solving i) and ii) we get

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

Answer:

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

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

Answer:

Question 27: Find the sum of odd integers from

Answer:

The series will be

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

Answer:

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:

… … … … … i)

Similarly, … … … … … ii)

Substituting in i)

Question 30: If be the sum of 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:

series is

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

Answer:

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

Answer:

Let for first A.P.

Let for second A.P.

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

Answer:

Let for first A.P.

Let for second A.P.