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.