Sequences that increase at increasing rates are sometimes described as “growing exponentially”. However, this is not always a correct use of the word. For example, both quadratic (constant second difference) and exponential growth share this property.

Question

Sequences that increase at increasing rates are sometimes described as “growing exponentially”. However, this is not always a correct use of the word. For example, both quadratic (constant second difference) and exponential growth share this property.

Suppose that a sequence begins 1, 2, 4. . .
a. What are the first ten terms of the sequence if it grows exponentially? b. What are the first ten terms of the sequence if it grows quadratically?
c. Compute the ratio of the 10th terms of the two sequences

Answer 1

The sequences are

1,2,4,8,16,32,64,128,256,512,1024
1,2,4,9,16,25,36,49,64,81,100

1024/100 = 256/25 = (16/5)^2

Answer 2

To determine the terms of the sequence, we need to understand the patterns for exponential and quadratic growth.

a. Exponential Growth:
In an exponential growth sequence, each term is obtained by multiplying the previous term by a constant value called the common ratio (r). In this case, the sequence starts with 1, and each subsequent term is obtained by multiplying the previous term by 2. So the first ten terms would be:

Term 1: 1
Term 2: 1 * 2 = 2
Term 3: 2 * 2 = 4
Term 4: 4 * 2 = 8
Term 5: 8 * 2 = 16
Term 6: 16 * 2 = 32
Term 7: 32 * 2 = 64
Term 8: 64 * 2 = 128
Term 9: 128 * 2 = 256
Term 10: 256 * 2 = 512

So, the first ten terms of the sequence in case of exponential growth are: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512.

b. Quadratic Growth:
In a quadratic growth sequence, each term is obtained by adding a constant value to the previous term. However, the difference between consecutive terms remains constant. In this case, the sequence starts with 1, and the difference between terms is 2. So, the first ten terms would be:

Term 1: 1
Term 2: 1 + 2 = 3
Term 3: 3 + 2 = 5
Term 4: 5 + 2 = 7
Term 5: 7 + 2 = 9
Term 6: 9 + 2 = 11
Term 7: 11 + 2 = 13
Term 8: 13 + 2 = 15
Term 9: 15 + 2 = 17
Term 10: 17 + 2 = 19

So, the first ten terms of the sequence in case of quadratic growth are: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.

c. Ratio of 10th Terms:
To compute the ratio of the 10th terms of the two sequences, we can simply divide the 10th term of the sequence with exponential growth by the 10th term of the sequence with quadratic growth.

Ratio = 512 / 19 ≈ 26.95

Therefore, the ratio of the 10th terms of the two sequences is approximately 26.95.