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 = (\frac{2\times3}{2})
1+2+3 = (\frac{3\times4}{2})
1+2+3+4 = (\frac{4\times5}{2})
1+2+3+4+5 = (\frac{5\times6}{2})
...
the next line in the sequence is 1+2+3+4+5+6 = (square) (do not simplify. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Identify the pattern

Looking at the given sequences:

• \(1 + 2=\frac{2\times3}{2}\)
• \(1 + 2+ 3=\frac{3\times4}{2}\)
• \(1 + 2+ 3+ 4=\frac{4\times5}{2}\)
• \(1 + 2+ 3+ 4+ 5=\frac{5\times6}{2}\)

The pattern is that the sum of the first \(n\) positive integers \(1 + 2+\cdots+n=\frac{n\times(n + 1)}{2}\)

Step2: Apply the pattern to the next term

For the sum \(1+2+3+4+5+6\), here \(n = 6\). Using the pattern \(\frac{n\times(n + 1)}{2}\), we substitute \(n=6\).

So the next line should be \(1 + 2+ 3+ 4+ 5+ 6=\frac{6\times7}{2}\)

Answer:

\(\frac{6\times7}{2}\)