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Complete the table and find the position of the term with value 50, given the pattern relates figure number (n) to the number of squares.

Mathematics

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2 parts
4.3

n2+1n^2 + 1

n×n+1n \times n + 1

4.4

7\text{7}

Step-by-step solution

3 steps

Here are the correct answers for 4.3 and 4.4:

Part 4.3

Complete the table

The pattern for the "Number of squares" for a given "Figure No" nn is n2+1n^2 + 1.

Here is the completed table:

Figure No12345nNumber of squares25101726n2+1Equivalent expressions1×1+12×2+13×3+14×4+15×5+1n×n+1Equivalent expressions12+122+132+142+152+1n2+1\begin{array}{|c|c|c|c|c|c|c|} \hline \textbf{Figure No} & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{n} \\ \hline \textbf{Number of squares} & 2 & 5 & 10 & 17 & 26 & \boxed{n^2 + 1} \\ \hline \textbf{Equivalent expressions} & 1 \times 1 + 1 & 2 \times 2 + 1 & 3 \times 3 + 1 & 4 \times 4 + 1 & 5 \times 5 + 1 & \boxed{n \times n + 1} \\ \hline \textbf{Equivalent expressions} & 1^2 + 1 & 2^2 + 1 & 3^2 + 1 & 4^2 + 1 & 5^2 + 1 & \boxed{n^2 + 1} \\ \hline \end{array}

Part 4.4

7\text{7}

What is the value of the position of the term whose value is 50?

  1. Set up the equation using the pattern identified in 4.3.

    The number of squares is given by the formula n2+1n^2 + 1, where nn is the position (Figure No). We are given that the value (number of squares) is 50.

    n2+1=50n^2 + 1 = 50

  2. Solve for nn.

    Subtract 1 from both sides:

    n2=50−1n^2 = 50 - 1 n2=49n^2 = 49

  3. Take the square root of both sides.

    n=49n = \sqrt{49} n=7n = 7

    The value of the position of the term whose value is 50 is 7\boxed{\text{7}}.

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