In mathville, the price of pencils has skyrocketed in recent years. each year for the last seven years the price has increased, and the new price is the sum of the prices for the two previous years. last year a pencil cost 60 cents. how much does a pencil cost today? how much did a pencil cost seven years ago?

The statement recalls Fibonacci sequence.

In Fibonacci sequence, the first two numbers are 0 and 1, and from there the numbers are the sum of previous two numbers, so the series is:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ...

As you see the first two numbers are fixed and the next are calculated witht the rule of the addition of the two previous numbers.

The problem sayt that the since 7 years ago the prices followed the rule of being the sum of the prices of the two years before.

We are in year 7, and know that in year 6 the price was 60c.

So, next year (year 8)  the price will be 60 + price in year 7 

year 8 = year7 + year6 = year7 + 60

and you can go backward.

year7 = year6  + year 5 = 60 + year5

year6 = year5 + year4 = 60

year5 = year4+year3

year4 = year3 + year2 

year3 = year2 + year1

Before that you need to make an assumption.

Was the price before the year 1 constant or they were as per the two first numbers of Fibonacci series, which are 0 and 1?.

We just know how the prices skyrocketed since year 1.

I will make the problem in two ways:

First approach: year 1's price = x and previos year price = 0

year    price

1         x
2         x+ 0 = x
3         x+x =2x
4         2x+x = 3x
5         3x+2x = 5x
6         5x+3x =8x

In that year the price is 60 c.

Then, 8x = 60 => x = 60/8 = 7.5 c

=> Price next year = 8x + 5x = 13x = 13(7.5) = 97.5 c
=> Price seven years ago = x = 7.5 c

Second approach: year's 1 price = x and two prices in the two previous years are 0 and 1

year        price
1             x
2             x+1
3             2x+1
4             3x+2
5             5x + 3
6             8x + 5

Then 8x + 5 = 60 => 8x = 55 => x = 55/8 = 6.875

=> price next year = 13x + 8 = 13(6.875) + 8 = 97.375

=> price year 1 - 6.875

There is not practical difference between the two assumptions, but the statement should be clear regarding all the information and that include how are the prices before seven years ago to calculate the prices in the two first years of the last seven years.

    
 

-2x + 7 = 7

subtract 7 from both sides.

-2x = 0

divide both sides by -2.

x = 0

the value of x is equal to 0.