The sum of the firest 9 terms of an A.P is 72 and the sum of the next 9 terms is 71, find the A.P

so the sum of 9 is 72 and

the sum of 18 = 72+71=143

(9/2)(2a + 8d) = 72 ---> 2a+8d=16
(18/2)(2a + 17d) = 143 ---> 2a + 17d = 143/9
subtract them
9d = -1/9

in 2a + 8d = 16
2a -8/9 = 16
2a = 152/9
a= 76/9

AP is 76/9 , 75/9, 74/9 ....

check for sum of 9 terms
first = 76/9
last = 76/9 + 8(-1/9) = 68/9
sum(9) = (first+last)(9/2) = (76/9+68/9)(9/2) = 72
looking good!

Well, it seems like this arithmetic progression is in a bit of a hot mess. Let's see if we can cool things down and figure it out.

Let's call the first term of the AP "a" and the common difference "d".

The sum of the first 9 terms is given by the formula S_1 = (9/2)(2a + (9-1)d), which is equal to 72. Simplifying, we get 9a + 36d = 72.

The sum of the next 9 terms is given by the formula S_2 = (9/2)(2a + (9-1)d), which is equal to 71. Simplifying again, we get 9a + 45d = 71.

Now, we have a pair of equations to solve this riddle. Subtracting the second equation from the first, we get 9d = 1, which means that d is equal to 1/9.

Substituting this value back into the first equation, we get 9a + 36(1/9) = 72. Simplifying, we have 9a + 4 = 72, meaning 9a = 68. Therefore, a = 68/9.

So, our arithmetic progression is a sequence, starting with 68/9, where each term increases by 1/9.

Now let's hope this AP doesn't heat things up any further!

Let's represent the first term of the arithmetic progression (AP) as 'a' and the common difference as 'd'.

We are given that the sum of the first 9 terms of the AP is 72. The formula for the sum of the first 'n' terms of an AP is given by:

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

Substituting the given values, we get:

72 = (9/2) * (2a + 8d) ... (Equation 1)

Similarly, we are given that the sum of the next 9 terms of the AP is 71. The sum of the next 'n' terms can be obtained by adding another 'n' terms to the previous sum. Therefore, the sum of the next 9 terms is:

S'n = Sn + (9/2) * (2a + 8d)

Substituting the values from Equation 1, we have:

71 = 72 + (9/2) * (2a + 8d)
71 = 72 + 9a + 36d

Rearranging the terms, we get:

71 - 72 = 9a + 36d
-1 = 9a + 36d ... (Equation 2)

Now, we have two equations (Equation 1 and Equation 2) to solve simultaneously:

Equation 1: 72 = (9/2) * (2a + 8d)
Equation 2: -1 = 9a + 36d

Let's solve these equations:

Multiplying Equation 2 by (-8), we get:

8 = -72a - 288d ... (Equation 3)

Adding Equation 1 and Equation 3, we eliminate the term 'a' and get:

80 = -280d

Dividing both sides by -280, we find the value of 'd':

d = -1/35

Substituting this value into Equation 1, we can find 'a':

72 = (9/2) * (2a + 8 * (-1/35))
72 = (9/2) * (2a - 8/35)
72 = (9/2) * ((70a - 8)/35)
72 = (9 * (70a - 8))/(2 * 35)
72 = (9 * (70a - 8))/70
72 = (90a - 9*8)/70
72 = (90a - 72)/70

Cross multiplying, we get:

5040 = 90a - 72
90a = 5112
a = 56.8

Now we have the first term 'a' and the common difference 'd', so the AP is:

56.8, 56.2, 55.6, ...

Therefore, the arithmetic progression is 56.8, 56.2, 55.6, ...

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

Let's analyze the given information.

1. The sum of the first 9 terms:
We know that the sum of an A.P can be calculated using the formula:
Sum = (n/2) * (2a + (n-1)d)
In this case, the sum of the first 9 terms is given as 72, so we can write:
72 = (9/2) * (2a + 8d)

2. The sum of the next 9 terms:
We are given that the sum of the next 9 terms is 71. Since the next 9 terms come right after the first 9 terms, they will start from the term after the last term of the first 9 terms. Therefore, the 10th term will be (a + 9d), the 11th term will be (a + 10d), and so on.
Using the same sum formula, we can calculate the sum of the next 9 terms:
71 = (9/2) * (2(a + 9d) + 8d)

Now we have a system of two equations with two variables (a and d). We can solve this system to find their values.

Let's simplify the equations:
1) 72 = 9a + 36d
2) 71 = 9a + 27d + 4d

Subtracting equation 2 from equation 1, we get:
1) -1 = 9d - 4d
-1 = 5d
d = -1/5

Substituting the value of d into either equation, we can solve for a:
2) 71 = 9a + 27(-1/5) + 4(-1/5)
71 = 9a - 27/5 - 4/5
71 = 9a - 31/5
71 + 31/5 = 9a
(355 + 31)/5 = 9a
386/5 = 9a
a = (386/5) * (1/9)
a = 386/45
a = 8.578

Therefore, the first term (a) of the A.P is approximately 8.578 and the common difference (d) is approximately -1/5 or -0.2. Hence, the A.P is:
8.578, 8.378, 8.178, 7.978, 7.778, 7.578, 7.378, 7.178, 6.978, ...

How did you get 68/9