the sum of first 8 terms of an arithmetic progression is 156.the ratio of its 12th term to its 68th term is 1:5.calculate the first term and the fifteenth term.

please help this questions please.

sum of 8 terms is 156

(8/2)(2a + 7d) = 156
4(2a + 7d) = 156
2a + 7d = 39

(a+11d) : (a + 67d) = 1 : 5

(a+11d)/(a+67d) = 1/5
5a + 55d = a + 67d
4a = 12d
a = 3d

sub that into 2a+7d=39
6d + 7d = 39
13d=39
d=3 , then in a = 3d ---> a = 9

first term is 9
term 15 = a+14d = 9+14(3) = 51

check:
sum(8) = (8/2)(18 + 7(3)) = 4(39) = 156 , checks !
term12 = a+11d = 42
term 68 = a+67d = 210
and 42/210 = 1/5 , YEahh!

Better explain

How many terms has the A.P Whose first term is 15 and the last is 57 given that the common difference is 3

I haven't get your response yet please help me

Maths

Oh, sure! I'd be happy to help! But just to clarify, are you looking for the first term and 15th term of the arithmetic progression?

To find the first term and the fifteenth term of the arithmetic progression, we can use the formulas for the sum of the first n terms and the nth term of an arithmetic progression.

Given:
Sum of the first 8 terms = 156
Ratio of the 12th term to the 68th term = 1:5

Step 1: Finding the common difference (d):
To find the common difference (d), we need to use the formula for the sum of the first n terms:
Sum = (n/2) * (2a + (n-1) * d)

Plugging in the given values:
156 = (8/2) * (2a + (8-1) * d)

Simplifying the equation:
156 = 4a + 28d

Step 2: Finding the first term (a):
Now that we have the equation with two variables (a and d), we need another equation involving the ratio of the 12th term to the 68th term.

The formula to find the nth term of an arithmetic progression is:
An = a + (n - 1) * d

Using this formula, we can write the equation for the 12th term in terms of a and d:
12th term = a + (12-1) * d = a + 11d

Similarly, the 68th term can be expressed as:
68th term = a + (68-1) * d = a + 67d

Given that the ratio of the 12th term to the 68th term is 1:5, we can write the following equation:
(a + 11d) / (a + 67d) = 1/5

Step 3: Solving the equations:
We now have a system of equations:

156 = 4a + 28d ---- (Equation 1)
(a + 11d) / (a + 67d) = 1/5 ---- (Equation 2)

Solving this system of equations will give us the values of a and d.

Once we have the value of the common difference (d), we can find the first term by rearranging Equation 1 and solving for a:
a = (156 - 28d)/4

To find the fifteenth term, we can use the formula for the nth term:
15th term = a + (15-1) * d = a + 14d

By substituting the value of a from the first term formula, we can find the value of the fifteenth term:
15th term = [(156 - 28d)/4] + 14d

Therefore, to calculate the first term and the fifteenth term, you need to solve the system of equations (Equation 1 and Equation 2) and substitute the value of d into the formulas for a and the fifteenth term.