Question

Use inductive reasoning to predict the next line in the sequence of computations. Then use a calculator or perform the arithmetic by hand to determine whether your conjecture is correct. 1+2 = 2x3/2 1+2+3 = 3x4/2 1+2+3+5 = 4x5/2 1+2+3+4+5 = 5x6/2 The next line in the sequence is 1 +2 +3 +4 +5 + 6 = __ (Do not simplify. Use integers or fractions for any numbers in the expression.)

Answer 1

The next line in the sequence is 1 + 2 + 3 + 4 + 5 + 6 = 6 x 7 / 2, which equals 21. The pattern of computing the sum of the first n natural numbers is confirmed to be correct.

Step-by-step explanation:

The pattern shown in the calculations suggests that to find the sum of the first n natural numbers, we use the formula n(n + 1) / 2. So far, the sequence corresponds to the sums 1, 1+2, 1+2+3, and 1+2+3+4+5. Notice that there's a mistake in the sequence provided: the fourth line skips the number 4 in the sum. Assuming this is a typo and it should be 1+2+3+4, the pattern continues orderly.

Following this pattern, for the next line in the sequence with n = 6, the computation would be:

1 + 2 + 3 + 4 + 5 + 6 = 6 x 7 / 2

Using simple arithmetic, whether by hand or with a calculator:

6 x 7 = 42

42 / 2 = 21

Therefore, the conjecture is that 1 + 2 + 3 + 4 + 5 + 6 equals 21. Now let's verify by performing the addition:

1 + 2 + 3 + 4 + 5 + 6 = 21

The conjecture is correct; the sum of the first six natural numbers is indeed 21.