Question

Consider the following sequence of successive numbers of the 2k -th power: 1, 2^2k, 3^2k, 4 ^2k, 5 ^2k, ... Show that the difference between the numbers in this sequence is odd for all k ∈ N.

Answer 1

Given  : 2^2k, 3^2k, 4^2k, 5^2k, ...

To Find : Show that the difference between the numbers in this sequence is odd for all k ∈ N.

Solution:

2^2k, 3^2k, 4^2k, 5^2k, ...

Any even number raised to power any +ve integer  is always even

Any odd number raised to power any +ve integer  is always odd

Hence two consecutive numbers on sequence are

EVEN & ODD

Difference between two odd number = EVEN

Difference between two even number = EVEN

Difference between  ODD & EVEN number = ODD

Hence the difference between the numbers in this sequence is odd for all k ∈ N.

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