Write a recursive formula for the following sequences:
1) 1,6,11,16...
2) 4, 12, 36, 108...
3) A ball is dropped from an initial height of 10 feet. The maximum heights the ball bounces are as shown (10 ft, 6 ft, 3.6 ft, 2.16 ft)
a-write a recursive formula
b-write an explicit formula
*basically, I don't know how to write a recursive or explicit formula*
#1
clearly you have to add 5 each time, so
T(n+1) = T(n) + 5
explicitly, that means
T(n) = -4 + 5n
You can do the others similarly.
Thank you!
One more question: how do you change an explcit formula to a recursive formula and vice versa?
it's not always easy.
Thus, since you know that #1 adds 5 each time, you expect
T(n) = 5n
But, T(1) = 5*1=5, not 1, so adjust the starting term:
T(1) = 5*1-4
T(n) = 5n-4 or, -4+5n
Thanks!
To write a recursive formula for a sequence, you need to determine the relationship between each term and preceding terms. Let's consider each sequence one by one:
1) 1, 6, 11, 16...
To find the recursive formula for this sequence, we need to identify the pattern. Here, each term is obtained by adding 5 to the previous term. So, the recursive formula for this sequence can be written as:
a(n) = a(n-1) + 5
where a(n) represents the nth term of the sequence.
2) 4, 12, 36, 108...
To find the recursive formula for this sequence, we need to identify the pattern. Here, each term is obtained by multiplying the previous term by 3. So, the recursive formula for this sequence can be written as:
a(n) = 3 * a(n-1)
where a(n) represents the nth term of the sequence.
3) A ball is dropped from an initial height of 10 feet. The maximum heights the ball bounces are as shown (10 ft, 6 ft, 3.6 ft, 2.16 ft).
a) Recursive Formula:
To find the recursive formula, observe that each maximum height is obtained by multiplying the previous height by 0.6. So, the recursive formula for this sequence can be written as:
h(n) = h(n-1) * 0.6
where h(n) represents the maximum height of the ball after n bounces.
b) Explicit Formula:
To find the explicit formula, we need to consider the initial height and the common ratio. In this case, the initial height is 10 feet, and the common ratio is 0.6. Therefore, the explicit formula for this sequence can be expressed as:
h(n) = 10 * (0.6)^(n-1)
where h(n) represents the maximum height of the ball after n bounces.
I hope this explanation helps you understand how to write recursive and explicit formulas!