in an ap the difference is between the 8th and 4th term is 20 and the 8th term is one and the half times the fourth term what is the common difference
In an AP the difference between the 8th and 4th terms is 20and the 8th is one and half x the 4th term
(i) Find the common difference
(ii) Find the first term
a+7d=3/2
Well, to find the common difference, we need to analyze the difference between consecutive terms. So let's get serious and do some math!
Let's call the 4th term 'x'. According to the information given, the 8th term would be 1.5 times 'x'.
Now, we're told that the difference between the 8th and 4th terms is 20. So, the equation we can set up is:
1.5x - x = 20
Simplifying that equation, we find:
0.5x = 20
Now, dividing both sides by 0.5, we find:
x = 40
So, the 4th term ('x') is 40.
And since the common difference is the difference between consecutive terms, we can say that the common difference is:
x - (x - d) = 40 - (40 - d) = 40
Wait, I seem to have made a mistake. Oh well, looks like the clown in me got the best of my calculations! The common difference is actually unknown based on the information given. My bad, I'll try to do better next time!
To find the common difference in an arithmetic progression (AP), you can use the formula:
nth term = first term + (n - 1) * common difference
In this case, let's denote the first term as 'a' and the common difference as 'd'.
Given that the difference between the 8th and 4th term is 20, we can write the equation:
8th term - 4th term = 20
Using the formula, we can substitute the values:
(a + 7d) - (a + 3d) = 20
Simplifying, we get:
4d = 20
Now, we also know that the 8th term is one and a half times the fourth term. We can express this as:
8th term = 1.5 * 4th term
Again, using the formula, we can substitute these values:
(a + 7d) = 1.5 * (a + 3d)
Expanding and simplifying, we get:
a + 7d = 1.5a + 4.5d
0.5a = 3.5d
Now, we have two equations:
4d = 20
0.5a = 3.5d
Solving the first equation, we find:
d = 5
Substituting this value in the second equation:
0.5a = 3.5 * 5
0.5a = 17.5
a = 35
Therefore, the common difference in this arithmetic progression is 5.
since T8 = T4+4d, d=5
and, just for completeness,
a+7d = 3/2 (a+3d)
a+35 = 3/2 (a+15)
2a+70 = 3a+45
a=25
So, the sequence is
25 30 35 40 45 50 55 60 ...