the sum of the first 8 terms of an Approval ia 12 and the the sum of the first 16 term is 56.find the first 3 terms3 of the Ap
No ideas?
8/2 (2a+11d) = 12
16/2 (2a+15d) = 56
Solve for a and d, and then get
S3 = 3/2 (2a+2d)
a=-4 and d=1
To find the first 3 terms of the arithmetic progression (AP), we can use the formula for the sum of an AP.
Let's denote the first term as 'a' and the common difference as 'd'.
The sum of the first 8 terms of the AP can be expressed as:
S8 = 8/2 * (2a + (8-1)d) = 12
The sum of the first 16 terms of the AP can be expressed as:
S16 = 16/2 * (2a + (16-1)d) = 56
Now we have two equations with two unknowns. Let's solve these equations simultaneously:
From equation 1:
8/2 * (2a + 7d) = 12
4(2a + 7d) = 12
8a + 28d = 12
From equation 2:
16/2 * (2a + 15d) = 56
8(2a + 15d) = 56
16a + 120d = 56
Now we have a system of equations:
8a + 28d = 12
16a + 120d = 56
We can solve this system of equations using any method, such as substitution or elimination.
Let's use elimination by multiplying the first equation by 2:
16a + 56d = 24
16a + 120d = 56
Subtract the first equation from the second equation:
(16a + 120d) - (16a + 56d) = 56 - 24
64d = 32
d = 32/64
d = 1/2
Now substitute the value of d back into the first equation to find 'a':
8a + 28(1/2) = 12
8a + 14 = 12
8a = 12 - 14
8a = -2
a = -2/8
a = -1/4
Therefore, the first term (a) of the AP is -1/4 and the common difference (d) is 1/2.
To find the first 3 terms of the AP, we substitute the values of 'a' and 'd' into the AP formula:
First term (a) = -1/4
Common difference (d) = 1/2
To find the second term:
a2 = a + d
a2 = (-1/4) + (1/2)
a2 = 1/4
To find the third term:
a3 = a2 + d
a3 = (1/4) + (1/2)
a3 = 3/4
Therefore, the first three terms of the AP are:
a1 = -1/4
a2 = 1/4
a3 = 3/4
To find the first three terms of the arithmetic progression (AP), we can use the formula for the sum of the first N terms of an AP:
Sum of first N terms = (N/2) * (2a + (N-1)d)
Where:
- N is the number of terms
- a is the first term
- d is the common difference between terms
Given that the sum of the first 8 terms is 12 and the sum of the first 16 terms is 56, we can set up two equations using the above formula:
Equation 1: 12 = (8/2) * (2a + (8-1)d)
Equation 2: 56 = (16/2) * (2a + (16-1)d)
Let's solve these equations to find the values of a and d, which will help us determine the first three terms of the AP.
Solving Equation 1:
12 = 4(2a + 7d)
3 = 2a + 7d [Divide both sides by 4]
Solving Equation 2:
56 = 8(2a + 15d)
7 = 2a + 15d [Divide both sides by 8]
Now, we have a system of equations:
3 = 2a + 7d
7 = 2a + 15d
Subtracting the two equations eliminates the 'a' term:
7 - 3 = (2a + 15d) - (2a + 7d)
4 = 8d
d = 4/8
d = 0.5
Substituting the value of d in one of the equations:
3 = 2a + 7(0.5)
3 = 2a + 3.5
2a = 3 - 3.5
2a = -0.5
a = -0.5/2
a = -0.25
Now that we have found the values of a and d, we can find the first three terms of the AP:
First term (a) = -0.25
Common difference (d) = 0.5
First term of the AP = -0.25
Second term = first term + common difference = -0.25 + 0.5 = 0.25
Third term = second term + common difference = 0.25 + 0.5 = 0.75
Therefore, the first three terms of the AP are -0.25, 0.25, and 0.75.