Question

1. Find the sum of the series (IMAGE) . Show your work.

2. A supermarket display consists of boxes of cereal. The bottom row has 23 boxes. Each row has three fewer boxes than the row below it. The display has six rows.
(a) Write and use a function to determine how many boxes are in the top row. Show your work.
(b) Use the appropriate formula to determine the number of boxes in the entire display. Show your work.

3. The number of visitors to a website in the first week is 50. The number of visitors each week is double the number of visitors the previous week. What is the total number of visitors to the website in the first 8 wk? Show your work.
Answer:

Answer 1

QUESTION 1 ANSWER

The series is

`\sum_(n=1)^(15){ (2n-1)}`

So the nth term of this sequence is

`U_n=2n-1`

We generate some few terms of the sequence as follows:

When
`n=1`,

The first term of the sequence becomes,

`U_1=2(1)-1`

`U_1=1`

When
`n=2`,

`U_2=2(2)-1`

`U_2=3`

When
`n=3`,

`U_2=2(3)-1`

`U_2=5`

The constant difference is
`d=2`.

The sum of the series is given by,

`S_n=(n)/(2)(2a+(n-1)d)`

There are 15 terms in the series, therefore
`n=15`

`S_(15)=(15)/(2)(2(1)+(15-1)2)`

`S_(15)=(15)/(2)(2+14(2))`

`S_(15)=(15)/(2)(30)`

`S_(15)=15* 15`

`S_(15)=225`

ANSWER TO QUESTION 2

The bottom row has
`23` boxes. This means
`a=U_1=23`

a) Since each row has 3 fewer boxes
`d=-3`.

The nth row is given by,

`U_n=23+(n-1)*(-3)`

`U_n=23+-3n+3`

`U_n=23-3n`

The top row is the 6th row, meaning
`n=6`.

`U_6=23-3(6)`

`U_6=23-18`

`U_6=5`

b) The number of boxes in the entire display is given by

`S_n=(n)/(2) (a+l)`

Where
`a=23`, the first term(row) and
`l=5`, the last term(row).

Since there are 6 rows,

`S_6=(6)/(2) (23+5)`

`S_6=3 (28)`

`S_6=84`

ANSWER TO QUESTION 3

The number of visitors in the first week is
`a=5`.

Since the number of visitors each week is doubled the number of visitors in the subsequent weeks, the sequence is a geometric progression with a common ratio of
`r=2`.

The general term of a geometric sequence is

`U_n=ar^(n-1)`

So in the 8th week, we have

`U_8=5* 2^(8-1)`

`U_8=5* 2^(7)`

`U_8=5* 128`

`U_8=640`

Hence 640 people visited the website in the 8th week