Question

12. The recurrence relation for the sequence 1,3,7,15,31,63...) _​

Answer 1

Let us look at the difference between these numbers:

3 - 1 = 2

7 - 3 = 4

15 - 7 = 8

31 - 15 = 16

63 - 31 = 32

Now, if we were to analyse these differences:

2 = 2^1

4 = 2^2

8 = 2^3

16 = 2^4

32 = 2^5

The next number in the above series would be : 2^6 = 64

Hence, difference between 63 and the next term in this sequence should be 64.

Therefore, next term would be : 63 + 64 = 127

Answer 2

Concept:

A recurrence relation is a mathematical expression that represents a sequence based on a rule. It assists in determining the following term (next term) based on the previous term. We can simply calculate the next term in a series if we know the preceding terms.

Given:

the sequence 1,3,7,15,31,63...

To find:

recurrence relation for the sequence 1,3,7,15,31,63...

Solution:

From the given series, the pattern can be observed as

$1 + 2= 3\\3+ 4 = 7\\7+ 8= 15\\\\15+ 16= 31\\31+ 32= 63\\63+ 64= 127$

Therefore the next term will be 127.

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