Question

Use inductive reasoning to find the next term in this sequence 2,3,5,9,17. Then explain the rule for the pattern.

Answer 1

We have been given the sequence 2,3,5,9,17.

We can write the terms of this sequence as

$2=2^0+1\\ 3=2^1+1\\ 5=2^2+1\\ 9=2^3+1\\ 17=2^4+1$

From the above term we can see that for the first term we take exponent 0 on 2 and then add 1 .

For second term we take exponent 1 on 2 and then add 1 .

For third term we take exponent 2 on 2 and then add 1 .

Using this fact for the next term of the sequence i.e. 6th term, we can take exponent 5 on 2 and then add 1 .

Therefore, next term of the sequence is given by

$2^5+1\\ =32+1\\ =33$

Therefore, the next term is 33.

Using the above facts, the pattern is given by

$a_n=2^{n-1}+1$