the third term of an AP is 5 while the 7th term is 13,find the 30th term of the AP
a+2d = 5
a+6d = 13
d=2, so
a+29d = a+6d+23d = 13+(23*2) = 59
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To find the 30th term of an arithmetic progression (AP), we need to determine the common difference (d) first.
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.
To find the 30th term of an arithmetic progression (AP) when given the third and seventh terms, we need to determine the common difference (d) first.
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.