Long Answer TypeProve the following by using the principle of mathematical induction for all 
a + (a + d) + (a + 2d) + ...........+ [a + (n - 1)d] = 
Let P(n) : a + (a + d) + (a + 2d) + .............+ [a + (n - 1)d] = 
I. For n = 1,
∴ P(1) is true
II. Suppose the statement is true for n=m, 
∴ 
.... (i)
III. For n = m + 1,
or 
From (i),
∴ 
which is true
∴ P (m + 1) is true
∴ P (m) is true 
P(m + 1) is true
Hence by the principle of mathematical induction, P(n) is true for all 
Prove the following by using the principle of mathematical induction for all 
n (n + 1) (n + 5) is a multiple of 3.
Prove the following by using the principle of mathematical induction for all 
is a multiple of 27 for all 
Prove the following by using the principle of mathematical induction for all 
is divisible by 11.
Prove the following by using the principle of mathematical induction for all 
is divisible by 8.
Prove by mathematical induction that sum of cubes of three consecutive natural numbers is divisible by 9.
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