Write the first five terms of a sequence. Don’t make your sequence too simple. Write both an explicit formula and a recursive formula for a general term in the sequence. Explain in detail how you found both formulas.
There is an example provided, (do not use the numbers in the example).
let the first five terms of a sequence be 5, 10, 20, 40 and 80. these terms are taken from a geometric sequence with first term a15 and common ratio r=2 therefore we have a2=a1* r a3=a2 * r Therefore the recursive formula is an+1=2an, a1=5. and explicit formula is an=a1rn-1
First five terms of the example sequence:
1. 5
2. 10
3. 20
4. 40
5. 80
The explicit formula for the general term in this sequence is:
an = 5 * 2^(n-1)
To find the explicit formula, we first recognize that this sequence is a geometric sequence where each term is obtained by multiplying the previous term by 2. We know the first term is 5 and the common ratio is 2. So, we use the formula for the nth term of a geometric sequence: an = a1 * r^(n-1), where a1 is the first term and r is the common ratio. Substituting a1 = 5 and r = 2 into the formula, we get an = 5 * 2^(n-1).
The recursive formula for the general term in this sequence is:
an+1 = 2 * an, with a1 = 5
To find the recursive formula, we notice that each term is obtained by doubling the previous term. We have the first term a1 = 5, and each subsequent term can be found by multiplying the previous term by 2. This relationship is represented in the recursive formula an+1 = 2 * an.