the sum of the first 9 terms of an A.p is 72 and the sum of the next 4 terms is 71. find the A.p
solution
once is enough, please!
Why did the A.P feel so conflicted? Because it was torn between being a sum of 72 and a sum of 71! That's quite the dilemma. So let's break it down.
The sum of the first 9 terms is 72, so let's call the first term "a" and the common difference "d". That means we can set up the equation:
(9/2) * [2a + (9-1)d] = 72
Simplifying that, we get:
(9/2) * [2a + 8d] = 72
18a + 72d = 144
Divide this by 18 to solve for a:
a + 4d = 8
Now, the sum of the next 4 terms is 71. Since we're considering the next 4 terms, the term right after the 9th term will be (a + 9d) and the following terms will be (a + 10d), (a + 11d), and (a + 12d). So we can set up another equation:
(4/2) * [2a + ((4-1)d)] = 71
2 * [2a + 3d] = 71
4a + 6d = 71
Now we have a system of equations:
a + 4d = 8
4a + 6d = 71
Now you can solve this system to find the values of "a" and "d" for your A.P. Good luck!
Step 1: Finding the common difference (d)
To find the common difference (d) of an arithmetic progression (AP), we subtract any two consecutive terms.
Let's take the first term (a) as a1 and the common difference (d) as d.
So, the second term (a2) will be given by:
a2 = a1 + d
Similarly, the third term (a3) will be given by:
a3 = a2 + d
And so on...
Step 2: Finding the sum of the first 9 terms
The sum of the first 9 terms is given as 72.
Using the formula for the sum of n terms (Sn) of an arithmetic progression:
Sn = (n/2)(2a1 + (n-1)d)
Substituting the given values:
9/2 * (2a1 + 8d) = 72
Step 3: Finding the sum of the next 4 terms
The sum of the next 4 terms is given as 71.
Using the same formula for Sn:
Sn = (n/2)(2a1 + (n-1)d)
Substituting the given values:
4/2 * (2a1 + 3d) = 71
Step 4: Solving the equations
Simplifying the equations obtained in step 2 and step 3, we have:
9a1 + 36d = 144 ----(Equation 1)
2a1 + 3d = 71 ----(Equation 2)
We have two equations with two unknowns (a1 and d). We can solve these equations simultaneously to find the values of a1 and d.
Step 5: Solving the system of equations
Multiply Equation 2 by 6:
12a1 + 18d = 426 ----(Equation 3)
Multiply Equation 1 by 2:
18a1 + 72d = 288 ----(Equation 4)
Subtract Equation 3 from Equation 4:
(18a1 + 72d) - (12a1 + 18d) = 288 - 426
Simplifying:
6a1 + 54d = -138 ----(Equation 5)
Now, subtract Equation 2 from Equation 5:
(6a1 + 54d) - (2a1 + 3d) = -138 - 71
Simplifying:
4a1 + 51d = -209 ----(Equation 6)
Now we have two equations with two unknowns (a1 and d). We can solve this system of equations simultaneously to find the values of a1 and d.
Step 6: Solving the system of equations (Equation 5 and Equation 6)
Multiply Equation 5 by 4:
24a1 + 216d = -552 ----(Equation 7)
Multiply Equation 6 by 6:
24a1 + 306d = -1254 ----(Equation 8)
Subtract Equation 7 from Equation 8:
(24a1 + 306d) - (24a1 + 216d) = -1254 - (-552)
Simplifying:
90d = -702
Divide by 90 on both sides:
d = -702/90
Simplifying,
d = -39/5
Step 7: Calculating a1
Using Equation 2:
2a1 + 3d = 71
Substituting the value of d we obtained in Step 6:
2a1 + 3(-39/5) = 71
Simplifying,
2a1 - 117/5 = 71
Adding 117/5 to both sides:
2a1 = (71)(5/1) + 117/5
Simplifying,
2a1 = 355/1 + 117/5
Combining the fractions by finding a common denominator:
2a1 = (355)(5/5) + (117)(1/5)
Simplifying,
2a1 = 1775/5 + 117/5
Adding the fractions:
2a1 = 1892/5
Dividing by 2:
a1 = (1892/5) / (2/1)
Dividing fractions is multiplying the first fraction by the reciprocal of the second fraction:
a1 = (1892/5) * (1/2)
Simplifying,
a1 = 1892/10
Simplifying further,
a1 = 189.2
Step 8: Writing the AP
We have found the first term (a1) as 189.2 and the common difference (d) as -39/5.
Therefore, the arithmetic progression (AP) is:
189.2, 189.2 - 39/5, 189.2 - (2 * 39/5), 189.2 - (3 * 39/5), ...
Simplifying further, we can express it as decimals:
189.2, 180, 170.8, 161.6, ...
Therefore, the arithmetic progression is 189.2, 180, 170.8, 161.6, ...
To find the arithmetic progression (A.P), we need to determine the first term (a) and the common difference (d).
Step 1:
Let's calculate the sum of the first 9 terms:
The formula for the sum of an arithmetic progression is given by:
Sn = (n/2)[2a + (n-1)d]
where Sn is the sum of the first n terms, a is the first term, and d is the common difference.
Given:
Sn = 72 (sum of the first 9 terms)
n = 9
We can substitute these values into the formula and simplify the equation:
72 = (9/2)[2a + (9-1)d]
Step 2:
Let's calculate the sum of the next 4 terms:
Sn = (n/2)[2a + (n-1)d]
Given:
Sn = 71 (sum of the next 4 terms)
n = 4
Again, we can substitute these values into the formula and simplify the equation:
71 = (4/2)[2a + (4-1)d]
Step 3:
We now have two equations:
72 = (9/2)[2a + 8d] -- Equation 1
71 = (4/2)[2a + 3d] -- Equation 2
Step 4:
Let's solve this system of equations using a method called substitution.
From Equation 2, we can simplify it to:
71 = 2(2a + 3d)
Dividing both sides by 2:
35.5 = 2a + 3d -- Equation 3
Step 5:
Now, let's substitute Equation 3 into Equation 1:
72 = (9/2)[2a + 8d]
Expanding the terms:
72 = (9/2) * 2a + (9/2) * 8d
Canceling out the factors of 2:
72 = 9a + 36d -- Equation 4
Step 6:
Now we have a system of two equations with two variables:
Equation 3: 35.5 = 2a + 3d
Equation 4: 72 = 9a + 36d
To solve this system, eliminate one variable by multiplying Equation 3 by 3 and Equation 4 by 2:
3 * (35.5) = 3(2a + 3d) --> 106.5 = 6a + 9d -- Equation 5
2 * (72) = 2(9a + 36d) --> 144 = 18a + 72d -- Equation 6
Step 7:
By subtracting Equation 5 from Equation 6, we can eliminate variable 'a':
144 - 106.5 = 18a + 72d - 6a - 9d
37.5 = 12a + 63d -- Equation 7
Step 8:
Now, let's solve Equation 7 for 'a':
37.5 - 63d = 12a
(37.5 - 63d)/12 = a
3.125 - 5.25d = a -- Equation 8
Step 9:
We can substitute the value of 'a' from Equation 8 into Equation 3:
35.5 = 2(3.125 - 5.25d) + 3d
35.5 = 6.25 - 10.5d + 3d
Simplifying the equation:
35.5 = -7.5d + 6.25
29.25 = -7.5d
Dividing both sides by -7.5 to solve for 'd':
d = -3.9
Step 10:
Now that we have the value of 'd', we can substitute it back into Equation 8 to get 'a':
a = 3.125 - 5.25d
a = 3.125 - 5.25(-3.9)
a = 3.125 + 20.475
a = 23.6
Step 11:
Thus, the first term of the arithmetic progression (A.P) is 23.6, and the common difference is -3.9.
Therefore, the arithmetic progression (A.P) is:
23.6, 19.7, 15.8, 11.9, 8, 4.1, 0.2, -3.7, -7.6, -11.5, -15.4, -19.3.