find the sum to 20 terms of an A.P. whose 4th term is 7 and 7th term is 13
"4th term is 7" ---> a+3d = 7
"7th term is 13" ---> a+6d = 13
subtract them:
3d = 6
d = 2
sub back into my first equation to find a,
then use the Sum of Terms formula to find sum(20)
To find the sum of an arithmetic progression (A.P.), we need to know the first term, the common difference, and the number of terms. However, the problem only provides the values of the 4th term and the 7th term.
To solve this, we need to find the first term (a) and the common difference (d) using the given information.
Given:
4th term (a4) = 7
7th term (a7) = 13
Step 1: Find the common difference (d):
From the given information, we can write the following equations:
a4 = a + 3d (since the 4th term is a + 3d)
a7 = a + 6d (since the 7th term is a + 6d)
Subtracting the first equation from the second equation, we get:
a7 - a4 = 3d
13 - 7 = 3d
6 = 3d
Dividing both sides of the equation by 3, we get:
d = 2
Step 2: Find the first term (a):
Substitute the value of d found in Step 1 into either of the equations to solve for a.
Let's use the equation a4 = a + 3d:
7 = a + 3(2)
7 = a + 6
a = 7 - 6
a = 1
Therefore, the first term (a) is 1 and the common difference (d) is 2.
Step 3: Find the sum of the 20 terms using the formula for sum of an A.P.:
The formula for the sum of an A.P. is given by:
Sn = (n/2)(2a + (n-1)d)
where Sn is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms.
Substituting the values into the formula, we get:
S20 = (20/2)(2(1) + (20-1)(2))
S20 = 10(2 + 19(2))
S20 = 10(2 + 38)
S20 = 10(40)
S20 = 400
Therefore, the sum of the first 20 terms of the arithmetic progression is 400.
To find the sum of the first 20 terms of an Arithmetic Progression (A.P.), we need to know two things: the first term (a) and the common difference (d).
Given that the 4th term (a_4) is 7 and the 7th term (a_7) is 13, we can use these values to find the first term and the common difference.
Using the formula for the nth term of an A.P., we have:
a_n = a + (n-1)d
For the 4th term:
a_4 = a + (4-1)d = 7
For the 7th term:
a_7 = a + (7-1)d = 13
We have two equations with two variables (a and d). We can solve this system of equations to find the values of a and d.
Subtracting the equation for a_4 from the equation for a_7:
(a + 6d) - (a + 3d) = 13 - 7
3d = 6
d = 2
Substituting the value of d into the equation for a_4:
a + 3(2) = 7
a + 6 = 7
a = 1
Now that we know the first term (a = 1) and the common difference (d = 2), we can find the sum of the first 20 terms using the formula for the sum of an A.P.
The sum of the first n terms of an A.P. is given by:
S_n = (n/2)(2a + (n-1)d)
For n = 20:
S_20 = (20/2)(2(1) + (20-1)(2))
= 10(2 + 19(2))
= 10(2 + 38)
= 10(40)
= 400
Therefore, the sum of the first 20 terms of the A.P. is 400.