Each term in the sequence is one less than twice the previous term.

So, \(\displaystyle \small x=2\left ( 17\right )-1\) 

\(\displaystyle \small x=33\)

The series is defined by n² – 1 starting at n = 1. The sixth number in the series then equal to 6² – 1 = 35.

The pattern of the sequence is (x+1) * 2.

We have the first 5 terms, so we need terms 6 and 7:

(78+1) * 2 = 158

(158+1) * 2 = 318

3 + 8 + 18 +38 + 78 + 158 + 318 = 621

Notice that in the sequence 

\(\displaystyle \small 3 , 7 , 11 , 15 , ...\)

each term increases by \(\displaystyle \small 4\).

It is always good strategy when attempting to find a pattern in a sequence to examine the difference between each term.

We continue the pattern to find:

The \(\displaystyle \small 5th\) term is \(\displaystyle \small 19\)

The \(\displaystyle \small \small 6th\) term is \(\displaystyle \small \small 23\)

The \(\displaystyle \small \small \small 7th\) term is \(\displaystyle \small \small \small 27\)

It is useful to note that the sequence is defined by,

\(\displaystyle \small S(n) = 4n -1\)

where n is the number of any one term.

We can solve

\(\displaystyle \small \small S(7) = 4(7) -1 = 28 - 1 = 27\)

to find the \(\displaystyle \small 7th\) term.

To reconstruct an original price from a sale price, use:

Original Price – Original Price * Mark-down-percent = Sale Price, or

Original Price * (1 - Mark-down-percent) = Sale Price

To do a double mark-down problem, we must do this twice.  For the 10%:

Original Sale Price * (1 – 10%) = $207.27

Original Sale Price = $207.27/0.9 = $230.30.

For the pre-all-discount price,

Original Price * (1 – 30%) = $230.30

Original Price = $230.30/0.7 = $329.00.

10% of the day one price = 0.1(100) = $10.

Therefore the day two price = 100 - 10 = $90.

10% of the day two price = 0.1(90) = $9.

Therefore the day three price = 90 + 9 = $99.

Let us assume that the original purse is $100. The price after the first reduction is $80. After the second reduction the price is now $56. The difference between 100 and 56 is 44, giving 44% off.

Widget 1 costs $20.

Widget 2 is on sale for 40%($20) off, or $8 off, or $20 – $8 = $12.

Two widgets during the sale cost $20 + $12 = $32.

Two widgets at regular price cost $20 + $20 = $40.

The total amount saved during the sale is $40 – $32 = $8.

This is the savings for two widgets, so the savings for one widget is $8/2 = $4.

The answer is $174.37.

The dress originally cost $225 but when it went on sale for 75% off we multiply the sale cost by 0.75. We see that through the sale we save $168.75 makeing the new cost of the dress $56.25.

Now we take the new cost of the dress ($56.25) and multiply that by 0.10 to represent the 10% discount. From this we see we save an additional $5.63 making the final cost of the dress $50.63.

The total savings on the dress sum up to $174.37.

Buying the stove at a 20% discount would be $240. If one buys it at a sale of 10%, with another 10% off then the price would be $243, so the difference is $3

20% of 300 is 0.2 * 300 = 60 → 300 – 60 = 240

10% of 300 is 0.1 * 300 = 30 → 300 – 30 = 270

10% of 270 is 0.1 * 270 = 27 → 270 – 27 = 243

243 – 240 = 3