The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of AP
First tell us what an AP is.
Arithmetic Progression?
If you are talking about arithmetic progressions, this one has the properties you require:
-13 -8 -3 2 7 12 17 22 27 32 ...
To find the first three terms of an arithmetic progression (AP), we'll use the given information about the sum of certain terms.
Let's denote the first term of the AP as 'a' and the common difference as 'd'. The general formula for the nth term of an AP is given by:
nth term = a + (n - 1)d
Using this formula, we can express the sum of the 4th and 8th terms and the sum of the 6th and 10th terms in terms of 'a' and 'd'.
1. Sum of the 4th and 8th terms:
The 4th term would be a + (4 - 1)d = a + 3d
The 8th term would be a + (8 - 1)d = a + 7d
According to the given information, the sum of these two terms is 24, so we have:
(a + 3d) + (a + 7d) = 24
2a + 10d = 24 (equation 1)
2. Sum of the 6th and 10th terms:
The 6th term would be a + (6 - 1)d = a + 5d
The 10th term would be a + (10 - 1)d = a + 9d
According to the given information, the sum of these two terms is 44, so we have:
(a + 5d) + (a + 9d) = 44
2a + 14d = 44 (equation 2)
Now we have a system of equations (equation 1 and equation 2) with two variables (a and d). We can solve this system of equations to find the values of 'a' and 'd'.
By subtracting equation 1 from equation 2, we get:
(2a + 14d) - (2a + 10d) = 44 - 24
4d = 20
d = 5
Now we substitute the value of 'd' into one of the equations (equation 1 or equation 2) to solve for 'a'.
Using equation 1:
2a + 10(5) = 24
2a + 50 = 24
2a = 24 - 50
2a = -26
a = -13
Therefore, the first term (a) is -13 and the common difference (d) is 5. Now we can find the first three terms of the AP:
1st term: a = -13
2nd term: a + d = -13 + 5 = -8
3rd term: a + 2d = -13 + 2(5) = 2
So, the first three terms of the AP are -13, -8, and 2.
use your two formulas for the general term of an AP and the sum of terms of an AP
a+3d + a+7d = 24 ----> 2a + 10d = 24
and
6/2[2a + 5d] + 10/2[2a+9d] = 44
now solve these two equations in two unknowns