What is the sum of the even, positive integers less than 62?
We are summing up $2+4+6+\cdots+60$. Factoring out a 2 and simplifying, we have $2(1+2+3+\cdots+30)=2\cdot\frac{30\cdot31}{2}=\boxed{930}$.
930
Level 4
Algebra
Problem: What is the sum of the even, positive integers less than 62?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the sum of the even, positive integers less than 62, first identify the even integers in this range. The even integers less than 62 are 2, 4, 6, ..., 60. This sequence is an arithmetic series where the first term \( a = 2 \) and the last term \( l = 60 \). The common difference \( d = 2 \).

Next, determine the number of terms \( n \) in this series. The formula for the \( n \)-th term of an arithmetic series is given by:

\[
l = a + (n-1)d
\]

Substituting the known values:

\[
60 = 2 + (n-1) \cdot 2
\]

Solving for \( n \):

\[
60 - 2 = (n-1) \cdot 2
\]
\[
58 = (n-1) \cdot 2
\]
\[
n-1 = 29
\]
\[
n = 30
\]

There are 30 even integers less than 62. The sum \( S_n \) of the first \( n \) terms of an arithmetic series can be calculated using the formula:

\[
S_n = \frac{n}{2} \cdot (a + l)
\]

Substituting the values:

\[
S_{30} = \frac{30}{2} \cdot (2 + 60)
\]
\[
S_{30} = 15 \cdot 62
\]
\[
S_{30} = 930
\]

Thus, the final answer is 930.<|END_OF_LATENT|> The final answer is $930$.