Finding the 49-Square Structure in a Geometric Sequence

Question

Here is a series of structures made of squares whose side length is 1 cm.

Is it possible to have a structure in the series that has 49 squares? If so, what element of the series is it?

Video Solution

Solution Steps

00:00 Will there be a term with 49 squares? If so, at what position?
00:05 Let's count the squares in each term
00:29 We see that the number of squares equals the term's position in the sequence
00:38 Therefore we can conclude this is the sequence formula
00:47 We want to find if there is a term with 49 squares
00:52 We'll substitute in the formula and solve for N
00:55 When taking a square root there are always 2 solutions (positive and negative)
00:59 There is no negative position in a sequence, so the positive solution is correct
01:02 And this is the solution to the question

Step-by-Step Solution

To determine if it is possible to have a structure in the series with 49 squares, consider the sequence of numbers representing perfect squares: 1, 4, 9, 16, 25, 36, 49, etc. Here, each number represents the total number of 1 cm x 1 cm squares contained in an n×n n \times n square.

The number 49 is indeed a perfect square, since 49=72 49 = 7^2 . This means that a structure made up of 49 squares can indeed be represented as a 7×7 7 \times 7 square.

Therefore, it is possible to have a structure in the series that has 49 squares, and it corresponds to the 7th square in the sequence (since n=7 n = 7 ), and this matches answer choice Yes, 7 7 .

Answer

Yes, 7 7

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