Prove the following by using the principle of mathematical induction for all n ∈ N:

n (n + 1) (n + 5) is a multiple of 3.

Find the sum of integers from 1 to 100 that are divisible by 2 or 5.

If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.

A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?

The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.

How many terms of G.P. 3, 3², 3³, … are needed to give the sum 120?

Find the sum of all numbers between 200 and 400 which are divisible by 7.

Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.

Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.

A coin is tossed. If it shows a tail, we draw a ball from a box which contains 2 red and 3 black balls. If it shows head, we throw a die. Find the sample space for this experiment.

Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.

Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.

How many terms of G.P. 3, 3², 3³, … are needed to give the sum 120?

The sum of first three terms of a G.P. is  and their product is 1. Find the common ratio and the terms.

If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in (A×B).

The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.

Find the sum to n terms of the series 1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 + …

Find the sum to n terms of the series whose nth term is given by n (n + 1) (n + 4).