Answer:
Finding the Nth Term:
The nth term is a formula that can be used to calculate any term in a sequence. The formula for the nth term is often expressed as:
an = a1 + (n - 1)d
where an is the nth term, a1 is the first term, n is the number of the term, and d is the common difference between each term.
For example, consider the following sequence:
2, 5, 8, 11, 14, ...
To find the nth term of this sequence using the formula, we would first identify a1 as 2 and d as 3 (the difference between each term). Then we can substitute these values into the formula to find the nth term:
an = 2 + (n - 1)3
So, for example, to find the 7th term of the sequence, we would plug in n=7:
a7 = 2 + (7 - 1)3
a7 = 2 + 18
a7 = 20
Therefore, the 7th term of the sequence is 20.
Gradient of a Line:
The gradient of a line is a measure of how steep the line is. It is also known as the slope of the line. The formula for calculating the gradient/slope of a line is:
gradient/slope= (change in y)/(change in x)
This means that the gradient/slope is the ratio of the change in the y-values to the change in the x-values.
For example, consider the following line:
y = 2x + 1
To find the gradient/slope of this line, we can calculate the change in y and change in x between any two points on the line. Let's say we choose the points (0,1) and (2,5). Then:
change in y = 5 - 1 = 4
change in x = 2 - 0 = 2
So the gradient/slope of the line is:
gradient/slope= 4/2
gradient/slope= 2
Therefore, the gradient/slope of the line y = 2x + 1 is 2.
Area Notation and Area Interval Notation:
Area notation and area interval notation are methods used to express the area underneath a curve. In area notation, we use the symbol ∫ to indicate that we are finding the area under a curve. For example:
∫ f(x) dx
This expression represents the area under the curve of the function f(x) between the limits of integration.
In area interval notation, we express the limits of integration using square brackets. For example:
[A,B] ∫ f(x) dx
This expression represents the area under the curve of the function f(x) between the limits A and B.
For example, consider the function f(x) = x^2 between the limits of integration x=0 and x=2. Using area notation, we would write:
∫ x^2 dx
Using area interval notation, we would write:
[0,2] ∫ x^2 dx
Fractions as Exponents:
Fractions can be used as exponents to represent roots or powers. For example, if we have a number x and we want to find the square root of x, we can write it as:
x^(1/2)
This means "x to the power of one-half", which is equivalent to taking the square root of x.
Similarly, if we want to find the cube root of x, we can write it as:
x^(1/3)
This means "x to the power of one-third", which is equivalent to taking the cube root of x.
For example, consider the number 8. To find the cube root of 8, we can write it as:
8^(1/3)
Using a calculator, we can calculate that:
8^(1/3) ≈ 2.08
Therefore, the cube root of 8 is approximately 2.08.