Question

Prove the following by using the principle of mathematical induction for all

is divisible by 11.

Solution

Let P(n) : is divisible by 11**I. **For n = 1,

P(1) : is divisible by 11

10^{1} + 1 is divisible by 11 11 is divisible by 11

∴ P(1) is true**II. **Suppose the statement is true for n = m,

∴ P(m) : is divisible by 11.

...(i)**III. **For n = m + 1,

is divisible by 11. ...(ii)

Now, [By (i)]

=

where k' = 100k -

∴ is divisible by 11

P (m + 1) is true.

∴ P(m) is true P (m + 1) is true

Hence, by induction, P(n) is true for all

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.

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a + (a + d) + (a + 2d) + ...........+ [a + (n - 1)d] =

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