Solving the Series Sum: 1+2+...+31+...+2+1

The question asks for the value of the series: $1+2+3+...+30+31+30+29+...+3+2+1$

Understanding the Series Pattern

This specific series increases from 1 up to a maximum number ($n$) and then decreases back to 1. The structure is:

$1 + 2 + ... + (n-1) + n + (n-1) + ... + 2 + 1$

This sum can be viewed as the sum of the first $n$ natural numbers plus the sum of the first $n-1$ natural numbers.

• Sum of first $n$ numbers ($S_n$): $1 + 2 + ... + n = \frac{n(n+1)}{2}$
• Sum of first $n-1$ numbers ($S_{n-1}$): $1 + 2 + ... + (n-1) = \frac{(n-1)n}{2}$

The total sum ($S_{total}$) is $S_n + S_{n-1}$.

Deriving the Efficient Formula

Let's combine the sums:

$S_{total} = \frac{n(n+1)}{2} + \frac{(n-1)n}{2}$

$S_{total} = \frac{n^2+n + n^2-n}{2}$

$S_{total} = \frac{2n^2}{2}$

$S_{total} = n^2$

This simplifies the calculation significantly: the sum equals the square of the maximum number in the series.

Calculating the Specific Series Value

In the given series, the maximum number is $n=31$.

Using the formula $S_{total} = n^2$:

$Sum = 31^2$

$Sum = 31 \times 31$

$Sum = 961$

Final Answer

The value of the series $1+2+3+...+30+31+30+29+...+3+2+1$ is 961.