The 16th term of an Arithmetric Progression is four times the 36th term, and exceeds it by 32. Finda the numbers.

To find the numbers in the arithmetic progression, we need to find the first term (a) and the common difference (d).

Given:
The 16th term, denoted by T16, is four times the 36th term, T36, and exceeds it by 32.

Let's set up the equations to represent the given information:

Equation 1: T16 = 4 * T36
Equation 2: T16 = T36 + 32

Since T16 = T36 + 32, we can substitute this into Equation 1:

T36 + 32 = 4 * T36

Expanding the equation:

32 = 4 * T36 - T36
32 = 3 * T36

Simplifying:

T36 = 32 / 3

Now that we know T36, we can substitute it into Equation 2 to find T16:

T16 = T36 + 32
T16 = 32 / 3 + 32

Simplifying:

T16 = (32 + 32) / 3
T16 = 64 / 3

Therefore, the 16th term, T16, is 64 / 3.

To find the first term (a) and the common difference (d), we will use the formulas for the terms in an arithmetic progression:

T_n = a + (n - 1) * d

Since T16 = 64 / 3, we can substitute it into the formula:

64 / 3 = a + (16 - 1) * d
64 / 3 = a + 15 * d

We now have one equation with two variables. We need another equation to solve for a and d.

We can use the fact that T36 = 32 / 3 to form a second equation:

32 / 3 = a + (36 - 1) * d
32 / 3 = a + 35 * d

Now we have a system of equations:

64 / 3 = a + 15 * d
32 / 3 = a + 35 * d

To solve this system, we can multiply the second equation by 2 to eliminate a:

64 / 3 = a + 15 * d
64 / 3 = 2 * (a + 35 * d)

Simplifying:

64 / 3 = a + 15 * d
64 / 3 = 2a + 70 * d

Rearranging the equations:

a + 15 * d = 64 / 3 -- (Equation 3)
2a + 70 * d = 64 / 3 -- (Equation 4)

Now we can solve this system of linear equations to find the values of a and d.

To find the numbers in the arithmetic progression, we need to first determine the common difference (d) and the first term (a).

Let's follow these steps:

Step 1: Identify the given information:
- The 16th term (A16) is four times the 36th term (A36).
- The 16th term (A16) exceeds the 36th term (A36) by 32.

Step 2: Express the terms in terms of the common difference and the first term:
- The 16th term: A16 = a + 15d
- The 36th term: A36 = a + 35d

Step 3: Translate the given information into equations:
- A16 = 4A36 (the 16th term is four times the 36th term)
- A16 = A36 + 32 (the 16th term exceeds the 36th term by 32)

Step 4: Solve the equations simultaneously:
Substitute the expressions for A16 and A36 into the equations:
a + 15d = 4(a + 35d) (substituting A16 and A36)
a + 15d = 4a + 140d
3a = 125d (equation 1)

a + 15d = a + 35d + 32 (substituting A16 and 32)
15d = 35d + 32
-20d = 32
d = -1.6 (equation 2)

Now we have the value for d, which is -1.6.

Step 5: Find the first term (a):
Substitute the value of d into equation 1:
3a = 125d
3a = 125(-1.6)
3a = -200
a = -200/3
a ≈ -66.67

So, the first term of the arithmetic progression is approximately -66.67.

Step 6: Calculate the terms of the arithmetic progression:
Now that we have the common difference (d ≈ -1.6) and the first term (a ≈ -66.67), we can calculate any term in the arithmetic progression using the formula An = a + (n-1)d, where n is the term number.

The 16th term:
A16 = a + (16-1)d
A16 = -66.67 + 15(-1.6)
A16 = -66.67 - 24
A16 ≈ -90.67

The 36th term:
A36 = a + (36-1)d
A36 = -66.67 + 35(-1.6)
A36 = -66.67 - 56
A36 ≈ -122.67

Therefore, the 16th term is approximately -90.67, and the 36th term is approximately -122.67.