To find the sum of all even integers from 10 to 70, we can use the formula for the sum of an arithmetic series:

$S_n = \frac{n}{2} \times (a_1 + a_n)$

where:

• $n$ is the number of terms,
• $a_1$ is the first term,
• $a_n$ is the last term.

Step 1: Identify the first term, last term, and number of terms

• The first even integer in the range is $a_1 = 10$.
• The last even integer in the range is $a_n = 70$.
• The difference between consecutive terms (common difference) is $d = 2$.

Step 2: Find the number of terms $n$

To find $n$, use the formula for the $n$-th term of an arithmetic sequence:

$a_n = a_1 + (n-1) \times d$

Substitute the values:

$70 = 10 + (n-1) \times 2$

Solving for $n$:

$70 = 10 + 2n - 2$ $70 = 8 + 2n$ $62 = 2n$ $n = \frac{62}{2} = 31$

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