if the 8th term of an AP is 37 and the 5th term is 15 more than the 12th term. find the AP. hence find the sum of the first 15 terms of the AP.

a+7d = 37

a+4d - (a+11d) = 15
-5d = 15
d = -3

into the first:
a - 21 = 37
a = 58

sum(15) = (15/2)(116 + 14(-3))
= 555

To find the arithmetic progression (AP), we need two pieces of information: the common difference (d) and the first term (a1).

Let's assume that the first term of the AP is "a1" and the common difference is "d".

Given that the 8th term of the AP is 37, we can write the equation: a1 + 7d = 37 ....(1)

Also, the 5th term is 15 more than the 12th term. So, we can write the equation: a1 + 4d = (a1 + 11d) + 15 ....(2)

Simplifying equation (2):
a1 + 4d = a1 + 11d + 15
-7d = 15
d = -15/7

Substituting the value of d into equation (1):
a1 + 7(-15/7) = 37
a1 - 15 = 37
a1 = 37 + 15
a1 = 52

Therefore, the first term (a1) is 52 and the common difference (d) is -15/7.

The AP is: 52, 37.857, 23.714, 9.571, -4.571, -18.714, -32.857, -47

To find the sum of the first 15 terms of the AP, we can use the formula for the sum of an AP:

Sn = (n/2) * (2a1 + (n-1)d)

Plugging in the values, we get:
S15 = (15/2) * (2(52) + (15-1)(-15/7))
S15 = (15/2) * (104 + 14*(-15/7))
S15 = (15/2) * (104 - 30/7)
S15 = (15/2) * (728/7 - 30/7)
S15 = (15/2) * (698/7)
S15 = (15/2) * 99.714
S15 = 748.71

Therefore, the sum of the first 15 terms of the given AP is 748.71.

To solve this problem, we need to find the common difference (d) of the arithmetic progression (AP) first.

We are given that the 8th term (a₈) of the AP is 37. The formula for the n-th term of an AP is given by: aₙ = a₁ + (n - 1) × d, where a₁ is the first term of the AP, and n is the position of the term in the AP.

Using this formula, we can write the equation for the 8th term as: a₈ = a₁ + 7d.

Substituting the given value, we have: 37 = a₁ + 7d.

Next, we are given that the 5th term (a₅) is 15 more than the 12th term (a₁₂). So, we can write the equation: a₅ = a₁₂ + 15.

Using the formula for the n-th term again, we can write these equations as:
a₁ + 4d = a₁ + 11d + 15.

Simplifying this equation, we get: 4d = 11d + 15.

Moving the terms around, we have: 7d = -15.

Dividing both sides of the equation by 7, we find: d = -15/7.

Now that we know the common difference (d) is -15/7, we can substitute this value back into either equation to find the first term (a₁).

Using the equation 37 = a₁ + 7d, we have: 37 = a₁ + 7(-15/7).

Simplifying further, we get: 37 = a₁ - 15.

Bringing the terms together, we find: a₁ = 52.

So, the first term (a₁) of the AP is 52, and the common difference (d) is -15/7.

To find the sum of the first 15 terms of an arithmetic progression, we can use the formula: Sₙ = (n/2) × (2a₁ + (n - 1) × d), where Sₙ represents the sum of the first n terms.

Substituting the given values, we have: S₁₅ = (15/2) × [2(52) + (15 - 1)(-15/7)].

Simplifying and evaluating this expression, we can find the sum of the first 15 terms of the AP.