Question

What is the sum of the sequence:
1+3+5+7+...1001

Answer 1

It is an arithmetic series.

The difference between consecutive terms, d, is 2

The first term, A1 = 1

The last term, An = 1001.

The formula to find the sum, S, is: S = n* (A1 + An)/2

n is the number of terms, which is [1001 + 1] / 2 = 501

Then, S = 501*(1 + 1001) / 2 = 251001

Answer 2

The sum of the sequence 1+3+5+7+...1001 is 251001.

Explanation:

Given sequence is 1 + 3 + 5 + 7 + ....+ 1001.

Here,
`a_0=1,a_1=3,a_2=5` and so on upto
`a_n=1001`

Lets find the difference between each term

`a_1-a_0=3-1=2`

`a_3-a_2=5-3=2`

`a_4-a_3=7-5=2`

We see that the difference between each term of the given sequence is 2 . Thus, it is an Arithmetic sequence.

Since we have to find the sum of the sequence

Sum of sequence of a given Arithmetic sequence is given as :

`S_n=(a_0+a_n)* (n)/(2)` .............(1)

But, first find the number of terms,

`a_n=a_0+(n-1)d`

Put values, we get,

`\Rightarrow 1001=1+(n-1)2`

`\Rightarrow 1001=1+2n-2`

`\Rightarrow 1001=2n-1`

`\Rightarrow 1001+1=2n`

`\Rightarrow 501=n`

Now put values,
`a_n=1001` ,
`501=n` and
`a_0=1`

in (1), we get,

`S_n=(a_0+a_n)* (n)/(2)`

`\Rightarrow S_n=(1+1001)* (501)/(2)`

`\Rightarrow S_n=(1002)* (501)/(2)`

`\Rightarrow S_n=251001`

Thus, the sum of the sequence 1+3+5+7+...1001 is 251001.