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

Answer:

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

Step-by-step 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 :

$S_n=(a_0+a_n)\times \frac{n}{2}$  .............(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,

$S_n=(a_0+a_n)\times \frac{n}{2}$

$\Rightarrow S_n=(1+1001)\times \frac{501}{2}$

$\Rightarrow S_n=(1002)\times \frac{501}{2}$

$\Rightarrow S_n=251001$

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