In an ap d sum of d 1st ten terms is 50 nd d fifth term is 3times the 2nd term.find the first nd d sum of d 1st 20 terms
Re-type in English please.
10a + 45d =50
.a +4d = 3a + 3d =a =d/2
10(d/2) + 45d = 50. .d=1 . a=1/2. . .
S20=20(1/2) +190(1) =200. . Ans =200
To find the first term of the arithmetic progression (AP), we can use the formula:
first term (a) = fifth term (5th term) - (4 * common difference)
Given that the fifth term (5th term) is 3 times the second term (2nd term), we can write:
5th term (tn) = 3 * 2nd term (a + d) ----(1)
We are also given that the sum of the first ten terms (Sn) is 50. The sum of an AP can be calculated using the formula:
Sum of n terms (Sn) = (n/2)(2a + (n-1)d)
Substituting the values given, we have:
50 = (10/2)(2a + (10-1)d)
Simplifying further:
50 = 5(2a + 9d)
10 = 2a + 9d ----(2)
Now we have a system of equations (equations 1 and 2) to solve for the first term (a) and the common difference (d).
Solving equation (1) and equation (2) simultaneously:
3 * 2nd term = 2a + 9d ----(3)
2a + 9d = 10 ----(4)
From equation (3):
2a = 3 * 2nd term - 9d
Substituting this value for 2a into equation (4):
3 * 2nd term - 9d + 9d = 10
3 * 2nd term = 10
2nd term (a + d) = 10/3
Now, we know the value of the 2nd term, we can substitute it back into Equation 1 to find the 5th term (tn):
5th term = 3 * (10/3) = 10
So, the 5th term (tn) is 10.
To find the first term (a), using equation (2):
10 = 2a + 9d
Substituting the value of the 5th term (10) and the 2nd term (10/3) into equation (2), we can solve for the first term:
10 = 2a + 9d
10 = 2a + 9(10/3)
10 = 2a + 90/3
10 = 2a + 30
2a = 10 - 30
2a = -20
a = -20/2
a = -10
So, the first term (a) is -10.
Now that we have the first term (a) and the common difference (d), we can find the sum of the first 20 terms (S20):
Using the formula for the sum of n terms:
Sum of 20 terms (S20) = (20/2)(2 * first term + (20-1) * common difference)
Substituting the values:
S20 = (20/2)(2 * -10 + (20-1) * 10)
Simplifying:
S20 = (10)(-20 + 19 * 10)
S20 = (10)(-20 + 190)
S20 = (10)(170)
S20 = 1700
Therefore, the sum of the first 20 terms (S20) is 1700.