Answer:
[tex]\sum _{i=2}^73i-15=-9[/tex]
Step-by-step explanation:
Given : [tex]\sum _{i=2}^73i-15[/tex]
We have to evaluate the given summation.
Consider [tex]\sum _{i=2}^73i-15[/tex]
using, we have,
[tex]\sum _{k=m}^n\:=\:\sum _{k=1}^n\:-\:\sum _{k=1}^{m-1}[/tex]
[tex]=\sum _{i=1}^73i-15-\sum _{i=1}^13i-15[/tex] ..............(1)
Consider, [tex]\sum _{i=1}^73i-15[/tex]
Apply sum rule,
[tex]\quad \sum a_n+b_n=\sum a_n+\sum b_n[/tex] , we have,
[tex]=\sum _{i=1}^73i-\sum _{i=1}^715[/tex]
Now, first consider, [tex]\sum _{i=1}^73i[/tex]
Apply constant multiplication rule,[tex]\quad \sum c\cdot a_n=c\cdot \sum a_n[/tex] we have,
[tex]=3\cdot \sum \:_{i=1}^7i[/tex]
[tex]=3\cdot \:28\\\\=84[/tex]
Also, [tex]\sum _{i=1}^715=105[/tex]
Thus, [tex]\sum _{i=1}^73i-15=84-105=-21[/tex]
Similarly, [tex]\sum _{i=1}^13i-15=-12[/tex]
Substitute in (1) , we have,
Thus, [tex]\sum _{i=2}^73i-15=-21-(-12)=-21+12=-9[/tex]
Thus, [tex]\sum _{i=2}^73i-15=-9[/tex]
