# 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

## 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.