Link to the diagram -
webactivities.files.wordpress.com/2010/09/figb1.jpg
a) Determine what kind of growth this is and explain how you know.
b) Explain how you can go about finding an expression for the number of small squares in the nth term.
c)What is the expression for the term of this sequence
4)Test your expression by showing that it gives the correct value for the 1
st,2nd and third terms.
The sequence of rectangles goes like
2*1, 3*3, 4*5, 5*7
so the formula for the nth term is (n+1)(2n-1) which is quadratic growth.
From "Please Help" to "Jack Will" to "Gary" all for the same question.
How can we keep you straight?
Why did you not describe the diagram or state the numbers 2, 9, 20, 35
in the first place, the pattern would have been easy to see, and avoided
unnecessary work by other tutors.
Did you notice that oobleck gave you the same answer you received 3 posts ago?
https://www.jiskha.com/questions/1828343/2-9-20-1-determine-what-kind-of-growth-this-is-and-explain-how-you-know-2-explain-how
https://www.jiskha.com/questions/1828355/2-9-20-1-determine-what-kind-of-growth-this-is-and-explain-how-you-know-2-explain-how
And here where I tested the formula as requested
https://www.jiskha.com/questions/1828363/1-2-2-9-3-20-a-determine-what-kind-of-growth-this-is-and-explain-how-you
a) To determine the kind of growth in the diagram, we need to observe the pattern of the figure and determine if it follows a specific growth pattern. Looking at the image in the provided link, we can see that each new term is created by adding one additional row of squares to the previous term. This indicates that the growth pattern of this diagram is linear.
b) To find an expression for the number of small squares in the nth term, we can analyze the pattern. Notice that for each subsequent term, the number of small squares is equal to the sum of all the previous terms plus the number of small squares in the current row.
To express this mathematically, we can use the formula for the sum of an arithmetic series. Let's assume the first term of the sequence has 1 square. Then the expression for the number of small squares in the nth term can be written as:
n^2 + (n-1)^2 + (n-2)^2 + ... + 1^2
c) The expression for the term of this sequence can be simplified to:
1^2 + 2^2 + 3^2 + ... + n^2
This can be further simplified using the formula for the sum of squares of the first n natural numbers, which is:
n(n+1)(2n+1)/6
So, the expression for the term of this sequence is:
n(n+1)(2n+1)/6
4) Let's test the expression by finding the values for the first, second, and third terms.
For the 1st term (n = 1):
Plugging in n = 1 into the expression, we get:
1(1+1)(2(1)+1)/6 = 1(2)(3)/6 = 6/6 = 1
As we assumed, the 1st term is 1.
For the 2nd term (n = 2):
Plugging in n = 2 into the expression, we get:
2(2+1)(2(2)+1)/6 = 2(3)(5)/6 = 30/6 = 5
The 2nd term is 5.
For the 3rd term (n = 3):
Plugging in n = 3 into the expression, we get:
3(3+1)(2(3)+1)/6 = 3(4)(7)/6 = 84/6 = 14
The 3rd term is 14.
Therefore, our expression gives the correct values for the 1st, 2nd, and 3rd terms.