Question

Hello,May I please request help for the following question that is in the uploaded picture?

Answer 1

Given:

a)

We get the following values from the graph.

`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.`

b)

The given nth term of the sequence is

`a_n=28n+63`

Required:

`We\text{ need to find }\sum_{i\mathop{=}1}^8a_i.`

Step-by-step explanation:

a)

Expand the sum.

`\sum_{i\mathop{=}1}^8a_i=a_1+a_2+a_3+a_4+a_5+a_6+a_7+a_8`

`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.}`

`\sum_{i\mathop{=}1}^8a_i=100+120+145+170+200+220+260+300`

`\sum_{i\mathop{=}1}^8a_i=1515`

b)

`Substitute\text{ n =1 in the equation }a_n=28n+63\text{ to find }a_1.`
`a_1=28(1)+63=28+63=91`

`Substitute\text{ n =2 in the equation }a_n=28n+63\text{ to find }a_2.`

`a_2=28(2)+63=56+63=119`

`Substitute\text{ n =3 in the equation }a_n=28n+63\text{ to find }a_3.`

`a_3=28(3)+63=84+63=147`

`Substitute\text{ n =4 in the equation }a_n=28n+63\text{ to find }a_4.`

`a_4=28(4)+63=112+63=175`

`Substitute\text{ n =5 in the equation }a_n=28n+63\text{ to find }a_5.`

`a_5=28(5)+63=140+63=203`

`Substitute\text{ n =6 in the equation }a_n=28n+63\text{ to find }a_6.`

`a_6=28(6)+63=168+63=231`

`Substitute\text{ n =7 in the equation }a_n=28n+63\text{ to find }a_7.`

`a_7=28(7)+63=196+63=259`

`Substitute\text{ n =8 in the equation }a_n=28n+63\text{ to find }a_8.`

`a_8=28(8)+63=224+63=287`

Consider the summation.

`\sum_{i\mathop{=}1}^8a_i=a_1+a_2+a_3+a_4+a_5+a_6+a_7+a_8`

`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.}`

`\sum_{i\mathop{=}1}^8a_i=91+119+147+175+203+231+259+287`

`\sum_{i\mathop{=}1}^8a_i=1512`

The actual sum is 1515 which is greater than 1512

So this value is underestimated.

Final answer:

a)

`\sum_{i\mathop{=}1}^8a_i=1515`

b)

Underestimated.