#
Problem
of the Week

Problem
C and Solution

Fruit
Display

## Problem

Jigme placed five oranges in a row on a long plate. He then placed
one apple in each of the spaces in the row between two oranges.

Next, he placed one banana in each of the spaces between two fruits
already in the row.

He then repeated this procedure with pears, then peaches, and finally
with strawberries.

Determine the total number of fruits in the row.

## Solution

After placing \(5\) oranges in the
row, there are \(4\) spaces between the
fruits. So Jigme placed \(4\) apples in
the row. At this point, there are \(5+4=9\) fruits in the row.

Since there are now \(9\) fruits in
the row, there are \(8\) spaces between
the fruits. So Jigme placed \(8\)
bananas in the row. At this point, there are \(9+8=17\) fruits in the row.

Since there are now \(17\) fruits in
the row, there are \(16\) spaces
between the fruits. So Jigme placed \(16\) pears in the row. At this point, there
are \(17+16=33\) fruits in the row.

Since there are now \(33\) fruits in
the row, there are \(32\) spaces
between the fruits. So Jigme placed \(32\) peaches in the row. At this point,
there are \(33+32=65\) fruits in the
row.

Finally, since there are now \(65\)
fruits in the row, there are \(64\)
spaces between the fruits. So Jigme placed \(64\) strawberries in the row. At this
point, there are \(65+64=129\) fruits
in the row.

Thus, there are \(129\) fruits in
the row in total.

**Extension:**

You may have noticed a pattern in the total number of fruits after
each new fruit was added. If Jigme placed fruits in this way using \(n\) different fruits, there will be a total
of \(2^{n+1}+1\) fruits in the row. Can
you see why? Start by looking at the number of spaces between the
fruits.