The value of 1 + 2 + 3 + ... + 30 + 31 + 30 + 29+...+3 + 2 + 1 = ?

Correct option is A

Given:

sequence is:

$$1+2+3+\cdots +30+31+30+29+\cdots +3+2+1$$

Formula Used:

$$S_n = \frac n 2(a+l)$$​​

Where:

​$$S_n$$​ is the sum of the first $n$ terms,

$n$ is the number of terms,

$a$ is the first term,

$l$ is the last term.

​Solution:

The given sequence is:

$$1+2+3+\cdots +30+31+30+29+\cdots +3+2+1$$

The sum of the first $31$ terms $1+2+3+\cdots +31$ 

$n=31$ (the number of terms),

$a=1$ (the first term),

$l=31$ (the last term).

$$S_{31} = \frac{31}{2} \times (1 + 31) = \frac{31}{2} \times 32 = 496$$

the sum of last 30 terms;

a = 30 ,  l = 1 

So,  

​$$S_{30} = \frac{30}{2} \times (30 + 1) = \frac{30}{2} \times 31 = 465$$​​

​$$\text{Total sum} = S_{31} + S_{30} = 496 + 465 = \bf 961$$​​

30+29+⋯+130 + 29 + \first term ​$$S_{30} = \frac{30}{2} \times (1 + 30) = \frac{30}{2} \times 31 = 465$$Now, the total sum is the sum of the increasing and decreasing parts:Thus, the value of the sequence is: $$961$$​​​

961\boxed{961}
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