Question
Give an expression or formula for the number of squares \( (s) \) in the \( n \)th picture. INSTRUCTIONS: - The first part of the formula has been written for you. You must complete the formula in terms of \( n \). - You do not need to simplify the expression.
Ask by Best Jimenez. in South Africa
Mar 13,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The number of squares \( s \) in the \( n \)th picture is
\[
s = 1 + 2n(n - 1).
\]
Solution
1. Recognize that in the first picture there is \( 1 \) square.
2. In each subsequent picture, additional squares are added. The pattern given is that the number of squares added at step \( k \) (for \( k = 1,\, 2,\, \dots,\, n-1 \)) is \( 4k \).
3. Therefore, the total number of squares \( s \) in the \( n \)th picture can be written as
\[
s = 1 + \sum_{k=1}^{n-1} 4k.
\]
4. Since the sum of the first \( n-1 \) positive integers is
\[
\sum_{k=1}^{n-1} k = \frac{n(n-1)}{2},
\]
we substitute to get
\[
s = 1 + 4\left(\frac{n(n-1)}{2}\right).
\]
This is the required expression for the number of squares in the \( n \)th picture.
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Simplify this solution Bonus Knowledge
To find the number of squares \( s \) in the \( n \)th picture, we can observe the pattern of squares being added as \( n \) increases. The initial part of the formula typically is \( s = n^2 + \) … where we recognize that each \( n \)th picture is creating a new square layer containing \( n \) squares on each side. Therefore, the completed expression could look like: \[ s = n^2 + (n-1)^2 + (n-2)^2 + \ldots + 1^2 \] This gives the total number of squares by summing the squares of consecutive numbers that fit within the structure defined by \( n \).
