Given:
[tex]a_n\text{ represents the number of autism cases diagnosed in the United States, in thousands , n years after 2000.}[/tex]a)
We get the following values from the graph.
[tex]a_1=100,a_2=120,a_3=145,a_4=170,a_5=200,a_6=220,a_7=260,\text{ and }a_8=300.[/tex]b)
The given nth term of the sequence is
[tex]a_n=28n+63[/tex]Required:
[tex]We\text{ need to find }\sum_{i\mathop{=}1}^8a_i.[/tex]Explanation:
a)
Expand the sum.
[tex]\sum_{i\mathop{=}1}^8a_i=a_1+a_2+a_3+a_4+a_5+a_6+a_7+a_8[/tex][tex]Substitute\text{ }a_1=100,a_2=120,a_3=145,a_4=170,a_5=200,a_6=220,a_7=260,\text{ and }a_8=300\text{ in the equation.}[/tex][tex]\sum_{i\mathop{=}1}^8a_i=100+120+145+170+200+220+260+300[/tex][tex]\sum_{i\mathop{=}1}^8a_i=1515[/tex]b)
[tex]Substitute\text{ n =1 in the equation }a_n=28n+63\text{ to find }a_1.[/tex][tex]a_1=28(1)+63=28+63=91[/tex][tex]Substitute\text{ n =2 in the equation }a_n=28n+63\text{ to find }a_2.[/tex][tex]a_2=28(2)+63=56+63=119[/tex][tex]Substitute\text{ n =3 in the equation }a_n=28n+63\text{ to find }a_3.[/tex][tex]a_3=28(3)+63=84+63=147[/tex][tex]Substitute\text{ n =4 in the equation }a_n=28n+63\text{ to find }a_4.[/tex][tex]a_4=28(4)+63=112+63=175[/tex][tex]Substitute\text{ n =5 in the equation }a_n=28n+63\text{ to find }a_5.[/tex][tex]a_5=28(5)+63=140+63=203[/tex][tex]Substitute\text{ n =6 in the equation }a_n=28n+63\text{ to find }a_6.[/tex][tex]a_6=28(6)+63=168+63=231[/tex][tex]Substitute\text{ n =7 in the equation }a_n=28n+63\text{ to find }a_7.[/tex][tex]a_7=28(7)+63=196+63=259[/tex][tex]Substitute\text{ n =8 in the equation }a_n=28n+63\text{ to find }a_8.[/tex][tex]a_8=28(8)+63=224+63=287[/tex]Consider the summation.
[tex]\sum_{i\mathop{=}1}^8a_i=a_1+a_2+a_3+a_4+a_5+a_6+a_7+a_8[/tex][tex]Substitute\text{ }a_1=91,a_2=119,a_3=147,a_4=175,a_5=203,a_6=231,a_7=259,\text{ and }a_8=287\text{ in the equation.}[/tex][tex]\sum_{i\mathop{=}1}^8a_i=91+119+147+175+203+231+259+287[/tex][tex]\sum_{i\mathop{=}1}^8a_i=1512[/tex]The actual sum is 1515 which is greater than 1512
So this value is underestimated.
Final answer:
a)
[tex]\sum_{i\mathop{=}1}^8a_i=1515[/tex]b)
Underestimated.
