If the sum of 8th and 9th of an AP is 72 and 4th term is -6,find the common difference
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Using your data in the definitions of AP ...
a+7d + a+8d = 72
2a + 15d = 72
a + 3d = -6
a = -6 - 15d
sub that into the first equation:
2(-6-15d) + 15d = 72
-12 -30d + 15d = 72
-15d = 84
d = -84/15
To find the common difference of an arithmetic progression (AP), we can use the formula for the nth term of an AP:
an = a1 + (n-1)d
where an represents the nth term, a1 is the first term, n is the term number, and d is the common difference.
Given information:
Sum of the 8th and 9th terms: a8 + a9 = 72
4th term: a4 = -6
To find the common difference, we need to plug in the given values into the formula:
a8 = a1 + (8-1)d
a9 = a1 + (9-1)d
We now have two equations with two unknowns (a1 and d):
(a1 + 7d) + (a1 + 8d) = 72
2a1 + 15d = 72 ------(1)
a1 + 3d = -6 ------(2)
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method by multiplying equation (2) by 2:
2(a1 + 3d) = 2(-6)
2a1 + 6d = -12 ------(3)
Now subtract equation (3) from equation (1):
(2a1 + 15d) - (2a1 + 6d) = 72 - (-12)
9d = 72 + 12
9d = 84
Dividing both sides by 9:
d = 84/9
d = 9.33 (rounded to two decimal places)
Therefore, the common difference (d) of the arithmetic progression is approximately 9.33.
To find the common difference of an arithmetic progression (AP), we need to first find the value of any two terms in the AP.
Given that the 4th term is -6 and the sum of the 8th and 9th terms is 72, we can use this information to find the common difference.
Let's start by finding the value of the 8th term (denoted by 'a8').
We know that the 4th term (denoted by 'a4') is -6, and the formula for the nth term of an AP is given by:
an = a1 + (n - 1) * d
Here, 'a1' represents the first term of the AP, and 'd' is the common difference.
So, for the 4th term ('a4'), substituting the given values into the formula:
a4 = a1 + (4 - 1) * d
-6 = a1 + 3d
Next, let's find the sum of the 8th and 9th terms (denoted by 'a8' and 'a9' respectively).
We know that:
a8 + a9 = 72
Substituting the formula for the nth term into this equation:
(a1 + (8 - 1) * d) + (a1 + (9 - 1) * d) = 72
Now, we have two equations:
1. -6 = a1 + 3d
2. (2a1 + 16d) + (2a1 + 18d) = 72
Simplifying the second equation:
4a1 + 34d = 72
2a1 + 17d = 36
We can now solve these two equations simultaneously to find the values of 'a1' and 'd'.
Using any method such as substitution or elimination, you can solve these equations to find the values of 'a1' and 'd'. The values of 'a1' and 'd' will allow you to determine the common difference of the arithmetic progression.