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a+6d = 13
d=2, so
a+29d = a+6d+23d = 13+(23*2) = 59
Step 1: Find the common difference (d):
The third term (a3) is 5, and the seventh term (a7) is 13.
a3 = a1 + (3 - 1)d ----(1)
a7 = a1 + (7 - 1)d ----(2)
Subtracting Equation (1) from Equation (2) to eliminate a1:
a7 - a3 = (7 - 1)d - (3 - 1)d
13 - 5 = 6d - 2d
8 = 4d
Dividing both sides by 4:
2 = d
So, the common difference (d) is 2.
Step 2: Find the 30th term (a30):
We know that the first term (a1) is not given, so we need to calculate it using the given information.
Using Equation (1):
a3 = a1 + (3 - 1)d
5 = a1 + (2)(2)
5 = a1 + 4
a1 = 5 - 4
a1 = 1
Now that we have the first term (a1) and the common difference (d), we can find the 30th term (a30) using the formula:
an = a1 + (n - 1)d
Substituting the values:
a30 = 1 + (30 - 1)(2)
a30 = 1 + 29(2)
a30 = 1 + 58
a30 = 59
Therefore, the 30th term of the arithmetic progression is 59.
The formula to find the nth term of an AP is:
an = a1 + (n - 1)d
Where:
an = nth term
a1 = first term
d = common difference
n = position of the term
Given information:
a3 = 5
a7 = 13
Step 1: Finding the common difference (d)
To find d, we can use the formula:
a7 = a1 + (7 - 1)d
Substituting the known values:
13 = a1 + 6d
Step 2: Solving for a1
To find a1, we can use the formula:
a3 = a1 + (3 - 1)d
Substituting the known values:
5 = a1 + 2d
Step 3: Solving the equations
Now we have two equations with two variables. We can solve them simultaneously.
Equation 1: 13 = a1 + 6d
Equation 2: 5 = a1 + 2d
To eliminate a1, we subtract equation 2 from equation 1:
13 - 5 = 6d - 2d
8 = 4d
Dividing both sides by 4:
d = 2
Step 4: Finding the first term (a1)
Substituting the value of d into equation 2:
5 = a1 + 2(2)
5 = a1 + 4
a1 = 5 - 4
a1 = 1
Now we know that the first term (a1) is 1 and the common difference (d) is 2.
Step 5: Finding the 30th term (a30)
Using the formula with the known values:
a30 = a1 + (30 - 1)d
a30 = 1 + (29)(2)
a30 = 1 + 58
a30 = 59
Therefore, the 30th term of the arithmetic progression is 59.