Find the general term of the sequence (3,5,9,17,33) then, find the next term.
General term is given by (2^n)+1 where n=0,1,2,3,...
Yes! Thank you.
clever !
To find the general term of a sequence, we need to look for a pattern or relationship between the terms. In this case, let's look at the differences between consecutive terms:
5 - 3 = 2
9 - 5 = 4
17 - 9 = 8
33 - 17 = 16
Notice that the differences between consecutive terms are powers of 2: 2, 4, 8, 16. This suggests that the sequence may be increasing by multiplying each term by a power of 2 and then adding one.
To verify this, let's calculate the differences between the differences:
4 - 2 = 2
8 - 4 = 4
16 - 8 = 8
The differences are consistent, confirming our hypothesis. Therefore, we can express the general term of the sequence as:
An = 2^n + 1
where n represents the position of the term in the sequence.
To find the next term, substitute the value of n with 5 (since the last given term is at position 4):
A5 = 2^5 + 1
= 32 + 1
= 33
Therefore, the next term in the sequence is 33.
consider the sequence of powers of 2:
2,4,8,16,32, ...
Is it clear now?