Question 1: Evaluate the following:

Answer:

Question 2: If , find the value of .

Answer:

Applying the formula, when

Therefore

Question 3: If , find .

Answer:

Applying the formula, when

Therefore

Question 4: If , find .

Answer:

Applying the formula, when

Therefore

Question 5: If , find .

Answer:

Applying the formula, when

Therefore

Question 6: If , find .

Answer:

Applying the formula, when

Therefore

Question 7: If , find .

Answer:

Given

Applying the formula, when

Therefore

Question 8: If , find .

Answer:

Comparing LHS with RHS we get,

If and

If , then

Hence can be or .

Question 9: If , find .

Answer:

Question 10: If , find .

Answer:

Question 11: If , find .

Answer:

Question 12: If and are in AP, then find .

Answer:

and are in AP

or

Question 13: If , find .

Answer:

Given

Question 14: If , find .

Answer:

Given

Applying the formula, when

Therefore

Question 15: If , then find the value of

Answer:

Given

Question 16: Prove that the product of consecutive negative integers is divisible by

Answer:

Let negative integers be

Product of negative integers

Therefore product of consecutive negative integers is divisible by

Question 17: For all positive integers , show that

Answer:

Therefore LHS = RHS hence proved.

Question 18: Prove that:

Answer:

RHS. Hence proved.

Answer:

Question 20: Let and be positive integers such that . The prove the following:

Answer:

Therefore LHS RHS

RHS

RHS. Hence proved.