created 05/24/03; edited 11/09/2012, 06/10/2018

Many of your programs from previous chapters can modified by changing`while`

loops into the equivalent`for`

loops. Some of these exercises are repeats of previous ones.

A breeding group of 20 bighorn sheep is released in
a protected area in Colorado.
It is expected that with careful management the
number of sheep, `N`

, after `t`

years will be given by the formula:

N = 220/(1 + 10(0.83)^{t})

and that the sheep population will be able to maintain itself without further supervision once the population reaches a size of 80.

Write a program (using a `for`

loop)
that writes out the value of
`N`

for
`t`

starting at zero and going up to 25.
How many years must the sheep heard be
supervised?

Hint: don't calculate `(0.83)`

"from scratch" each time the formula is used.
Use a variable ^{t}`power`

that is multiplied by
0.83 in each iteration of the loop.
What value should it be initialized to?

(Problem from Howard Anton, *Calculus,*
6^{th} ed., p. 105. )

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Write a program that asks the user for an integer N and then calculates two things:
(a) the sum of all odd integers from 1 to N,
and (b) N^{2}.
Use a `for`

loop with appropriate increment.

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Write a program that asks the user for an integer N and then calculates
the sum of all integers from 1 to N that divide N with zero left over.
Use the `%`

operator.

For example, if N is 10, then the sum of divisors is 1+2+5 = 8. Notice that 1 is considered a divisor and than no integer greater than N/2 is a divisor of N.

Another example: if N is 6, then the sum of divisors is 1+2+3 = 6.
In this case, 6 is considered a *perfect number* because the sum of divisors is the number itself.

Include an `if`

statement that determines if N is a perfect number.

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Write a program that asks the user for an integer N and then determines if N is the sum of two the squares of two integers.

Integers that are a sum of two squares:

`4 = 0`

^{2}+ 1^{2}`5 = 2`

^{2}+ 1^{2}`10 = 3`

^{2}+ 1^{2}`25 = 3`

^{2}+ 4^{2}= 0^{2}+ 5^{2}

Integers that are not a sum of two squares:

`3`

`6`

`39`

`48`

One way to do this is with a doubly-nested loop that generates trial integers `a`

and `b`

,
squares each, and tests if the sum is equal to `N`

.
Of course, when `a`

exceeds ^{2}`N`

the outer loop is done.
Decide on a termination condition for the inner loop.

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Write a program that reads 5 integers from a file,
computes their sum
and their maximum and prints these values
to the monitor.
Do this by modifying the summing program
from the chapter.
Use an `int`

variable
called `max`

which is initialized to
the first value in the file.
This calls for an extra set of input statements
before the loop starts.
To compute the maximum use an `if`

statement nested inside the loop.

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