Maximum Sum of Arithmetic Progression

The sequence 44, 42, 40, ... represents an arithmetic progression (AP).

• The first term is $a = 44$.
• The common difference is $d = 42 - 44 = -2$.

The phrase "maximum sum" implies summing only the positive terms because the common difference is negative, causing the terms to decrease.

Identifying Positive Terms

We need to find the number of terms ($n$) that are positive. The formula for the $n$-th term ($a_n$) of an AP is:

$a_n = a + (n-1)d$

Substituting the values:

$a_n = 44 + (n-1)(-2)$

$a_n = 44 - 2n + 2$

$a_n = 46 - 2n$

To find when the terms are positive, we set $a_n > 0$:

$46 - 2n > 0$

$46 > 2n$

$23 > n$

The largest integer value for $n$ is 22. Therefore, there are 22 positive terms in the sequence.

The last positive term is $a_{22} = 46 - 2(22) = 46 - 44 = 2$.

Calculating the Sum

The maximum sum is the sum of these 22 positive terms ($S_{22}$). The formula for the sum of an AP is:

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

Plugging in the values for $n=22$, $a=44$, and $a_{22}=2$:

$S_{22} = \frac{22}{2}(44 + 2)$

$S_{22} = 11(46)$

$S_{22} = 506$

The maximum sum achievable by summing the positive terms of this sequence is 506.