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?

I immediately thougth in Fibonacci sequence:  the first two numbers are 0 and 1, and from there the numbers are the sum of previous two numbers.

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

The first two numbers are fixed and the next are calculated witht by adding up of the two previous numbers.

The problem tells that since 7 years ago, the prices followed the same rule.

Current year is 7 and in year 6 the price was 60c.

Next year, year 8,  the price will be 60 + price in year 7 

You need to make an assumption about how to start the sequence.

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 three ways:

First approach: year 1's price = x and previous 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

Third approach: prices before year 1, equal to the same price of year 1, x

year    price
0         x
1         x
2         x+x =2x
3         2x+x = 3x
4         3x+2x=5x
5         5x + 3x = 8x
6         8x + 5x = 13x

13x = 60 => x = 60 / 13 = 4.62

=> Next year: price =13x + 8x = 21x = 21(4.62) = 96.92

=> price seven years ago = 4.62