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, [tex]a_0=1,a_1=3,a_2=5[/tex] and so on upto [tex]a_n=1001[/tex]
Lets find the difference between each term
[tex]a_1-a_0=3-1=2[/tex]
[tex]a_3-a_2=5-3=2[/tex]
[tex]a_4-a_3=7-5=2[/tex]
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 :
[tex]S_n=(a_0+a_n)\times \frac{n}{2}[/tex] .............(1)
But, first find the number of terms,
[tex]a_n=a_0+(n-1)d[/tex]
Put values, we get,
[tex]\Rightarrow 1001=1+(n-1)2[/tex]
[tex]\Rightarrow 1001=1+2n-2[/tex]
[tex]\Rightarrow 1001=2n-1[/tex]
[tex]\Rightarrow 1001+1=2n[/tex]
[tex]\Rightarrow 501=n[/tex]
Now put values, [tex]a_n=1001[/tex] , [tex]501=n[/tex] and [tex]a_0=1[/tex]
in (1), we get,
[tex]S_n=(a_0+a_n)\times \frac{n}{2}[/tex]
[tex]\Rightarrow S_n=(1+1001)\times \frac{501}{2}[/tex]
[tex]\Rightarrow S_n=(1002)\times \frac{501}{2}[/tex]
[tex]\Rightarrow S_n=251001[/tex]
Thus, the sum of the sequence 1+3+5+7+...1001 is 251001.
