The sum of the given series [tex]\sum _{a=1}^{10}\:\left(2a+2\right)[/tex] is 130.
Step-by-step explanation:
The sum of series need to be found is [tex]\sum _{a=1}^{10}\:\left(2a+2\right)[/tex] .
Apply sum rule,
[tex]\sum x_n+y_n=\sum x_n+\sum y_n[/tex]
[tex]\sum _{a=1}^{10}\:\left(2a+2\right).[/tex] =[tex]\sum 2a+\sum 2[/tex].
Apply the constant multiplication rule,
[tex]\sum c\cdot a_n=c\cdot \sum a_n[/tex] .
[tex]\sum 2\cdot a=2\cdot \sum a[/tex].
[tex]\sum _{a=1}^{10}n[/tex] = 1+2+3+4+5+6+7+8+9+10.
[tex]\sum _{a=1}^{10}n[/tex] = 55.
[tex]\sum 2\cdot a[/tex]=2×55.
[tex]\sum 2\cdot a [/tex]=110.
To find [tex]\sum _{a=1}^{10}2[/tex],
Apply sum rule, [tex]\sum _{k=1}^n\:a\:=\:a\cdot n[/tex] ,
[tex]\sum _{a=1}^{10}2[/tex] =2×10.
[tex]\sum _{a=1}^{10}2[/tex] = 20.
[tex]\sum _{a=1}^{10}\:\left(2a+2\right)[/tex] = 110+20.
[tex]\sum _{a=1}^{10}\:\left(2a+2\right)[/tex] =130.
