Using inductive reasoning, what is the next bottom layer in the pattern: 1; 1+2; 1+3+3; 1+3+7+4; 1+3+7+15+5; ?

Using inductive reasoning, what is the next bottom layer in the pattern: 1; 1+2; 1+3+3; 1+3+7+4; 1+3+7+15+5; ?

• A. 1+3+7+15+31+63+7
• B. 1+2+4+8+16
• C. 1+3+7+15+31
• D. 1+3+7+15+31+63

Answer: (A)

The pattern adds numbers that are one less than powers of two on the left, while the last number in each row increases by 1. The left-hand terms go 1, 3, 7, 15, … which are 2^1−1, 2^2−1, 2^3−1, 2^4−1, and so on. To extend the pattern, you continue with the next two numbers in that left sequence: 2^5−1 = 31, 2^6−1 = 63, and then the final number (the row’s last element) increases to the next natural number, which gives 7. So the next bottom layer is 1+3+7+15+31+63+7.

The pattern adds numbers that are one less than powers of two on the left, while the last number in each row increases by 1. The left-hand terms go 1, 3, 7, 15, … which are 2^1−1, 2^2−1, 2^3−1, 2^4−1, and so on. To extend the pattern, you continue with the next two numbers in that left sequence: 2^5−1 = 31, 2^6−1 = 63, and then the final number (the row’s last element) increases to the next natural number, which gives 7. So the next bottom layer is 1+3+7+15+31+63+7.