Chapter 9.2, Problem 4E

Solution

a.

To write the next line of the given pattern.

  $1^2+1^2+2^2+3^2+5^2+8^2=104=8.13$

Given:

The given pattern is:

  $\begin{array}{l}{1}^2+1^2=2=1.2\\ 1^2+1^2+2^2=6=2.3\\ 1^2+1^2+2^2+3^2=15=3.5\\ 1^2+1^2+2^2+3^2+5^2=40=5.8\end{array}$

Calculation:

After observing the pattern, it is concluded that the new term is obtaining by taking the sum of $3\text{ and }5,$ i.e.

  $3+5=8$

Now, the next line of the series is:

  $\begin{array}{l}L:1^2+1^2+2^2+3^2+5^2+8^2\\ \Rightarrow L:1^2+1^2+2^2+3^2+5^2+8^2=1+1+4+9+25+64\\ \Rightarrow L:1^2+1^2+2^2+3^2+5^2+8^2=104\\ \Rightarrow L:1^2+1^2+2^2+3^2+5^2+8^2=8\cdot 13\end{array}$

Hence, the next line is:

  $1^2+1^2+2^2+3^2+5^2+8^2=104=8.13$

b.

To determine the sum of the squares of the first ten terms of the Fibonacci sequence.

Sum of squares of the first ten terms is equal to $4895.$

Calculation:

The term in the Fibonacci sequence is the sum of preceding two terms. The terms for the Fibonacci sequence:

  $\begin{array}{l}{1}^{st}\text{ term }=1\\ 2^{nd}\text{ term }=1\\ 3^{rd}\text{ term }=2\\ 4^{th}\text{ term }=3\\ 5^{th}\text{ term }=5\\ 6^{th}\text{ term }=8\end{array}$

The other successive terms are:

  $\begin{array}{l}{7}^{th}\text{ term }=8+5=13\\ 8^{th}\text{ term }=13+8=21\\ 9^{th}\text{ term }=21+13=34\\ {10}^{th}\text{ term }=34+21=55\\ {11}^{th}\text{ term }=55+34=89\end{array}$

Now, the sum of squares of the Fibonacci sequence of $n$ terms is given by:

  $\left[n^{th}term\right]\cdot \left[{\left(n+1\right)}^{th}term\right]$

Now, calculating the sum of the squares of the first ten terms as:

  $\begin{array}{l}S=\left[{10}^{th}term\right]\cdot \left[{\left(10+1\right)}^{th}term\right]\\ \Rightarrow S=\left[{10}^{th}term\right]\left[{11}^{th}term\right]\\ \Rightarrow S=\text{55}\times \text{89}\\ \Rightarrow S=\text{4895}\end{array}$

Hence, the sum of the squares of the first ten terms is equal to $4895.$

c.

To verify the prediction made in part (b).

The prediction is correct.

Given:

The first ten terms are given as:

  $1,1,2,3,5,8,13,21,34,55$

Calculation:

Now, calculating the sum of squares of first ten terms as:

  $\begin{array}{l}S=1^2+1^2+2^2+3^2+5^2+8^2+{13}^2+{21}^2+{34}^2+{55}^2\\ \Rightarrow S=1+1+4+9+25+64+169+441+1156+3025\\ \Rightarrow S=4895\end{array}$

Hence, the prediction is correct.