Chapter 3.6, Problem 28WE

a.

To determine

To calculate: the value for the given expression by substituting n = 1 to 9 .

Answer to Problem 28WE

  $n^2+n+11$ = 13, 17, 23, 31, 41, 53, 67, 83, 101 for n = 1 to 9

Explanation of Solution

Given information: $n^2+n+11$

Calculation :

For $n=1$ , $n^2+n+11=1^2+1+11=13$

  $n=2$ , $n^2+n+11=2^2+2+11=17$

  $n=3$ , $n^2+n+11=3^2+3+11=23$

  $n=4$ , $n^2+n+11=4^2+4+11=31$

  $n=5$ , $n^2+n+11=5^2+5+11=41$

  $n=6$ , $n^2+n+11=6^2+6+11=53$

  $n=7$ , $n^2+n+11=7^2+7+11=67$

  $n=8$ , $n^2+n+11=8^2+8+11=83$

  $n=9$ , $n^2+n+11=9^2+9+11=101$

b.

To determine

To identify: using inductive reasoning a suitable guess for the value of $n^2+n+11$ , for any positive value for n.

Answer to Problem 28WE

  $f\left(n\right)=f\left(n-1\right)+2n$

Explanation of Solution

Given information: $n^2+n+11$

From Sub-part (a):

n	1	2	3	4	5	6	7	8	9
$n^2+n+11$	13	17	23	31	41	53	67	83	101

  

From the above relationship, it can be guessed that the value for $f\left(n\right)=n^2+n+11$ can be given as:

  $f\left(n\right)=f\left(n-1\right)+2n$ , for $n\ge 2$

c.

To determine

To verify: whether the guess for the given expression is correct by substituting n = 10, 11

Explanation of Solution

Given information: $n^2+n+11$

Verification:

For

  $n=10$ , $n^2+n+11=100+10+11=121$

  $n=11$ , $n^2+n+11=121+11+11=143$

Using the guess: $f\left(n\right)=f\left(n-1\right)+2n$ for $n\ge 2$

For

  $n=10$ , $f\left(10\right)=f\left(9\right)+\left(2\times 10\right)=101+20=121$

  $n=11$

, $f\left(11\right)=f\left(10\right)+\left(2\times 11\right)=121+22=143$