Question

The sum of two consecutive integers is no more than 209.

What is the largest possible integer?

Answer 1

Let the two consecutive integers be \( n \) and \( n + 1 \). The sum of these integers can be expressed as:

\[
n + (n + 1) = 2n + 1
\]

According to the problem, this sum is no more than 209:

\[
2n + 1 \leq 209
\]

To solve for \( n \), first subtract 1 from both sides:

\[
2n \leq 208
\]

Next, divide both sides by 2:

\[
n \leq 104
\]

The largest integer \( n \) that satisfies this inequality is 104. The consecutive integer following \( n \) is:

\[
n + 1 = 104 + 1 = 105
\]

Thus, the largest possible integer among the two consecutive integers is:

\[
\boxed{105}
\]