Answer:
(a) 498501
(b) 251001
Step-by-step explanation:
According Gauss's approach, the sum of a series is
[tex]sum=\frac{n(a_1+a_n)}{2}[/tex] .... (1)
where, n is number of terms.
(a)
The given series is
1+2+3+4+...+998
here,
[tex]a_1=1[/tex]
[tex]a_n=998[/tex]
[tex]n=998[/tex]
Substitute [tex]a_1=1[/tex], [tex]a_n=998[/tex] and [tex]n=998[/tex] in equation (1).
[tex]sum=\frac{998(1+998)}{2}[/tex]
[tex]sum=499(999)[/tex]
[tex]sum=498501[/tex]
Therefore the sum of series is 498501.
(b)
The given series is
1+3+5+7+...+ 1001
The given series is the sum of dd natural numbers.
In 1001 natural numbers 500 are even numbers and 501 are odd number because alternative numbers are even.
[tex]a_1=1[/tex]
[tex]a_n=1001[/tex]
[tex]n=501[/tex]
Substitute [tex]a_1=1[/tex], [tex]a_n=1001[/tex] and [tex]n=501[/tex] in equation (1).
[tex]sum=\frac{501(1+1001)}{2}[/tex]
[tex]sum=\frac{501(1002)}{2}[/tex]
[tex]sum=501(501)[/tex]
[tex]sum=251001[/tex]
Therefore the sum of series is 251001.
