If the 9th term of an A.P is five times the 5th term. Find the relationship between a and d?
Nth = a (n -1)d
¡à 9th = a (9 - 1)d
9th = a 8d
5th = a (5 -1)d
5th = a 4d
9th term of an A.P is five times the 5th term
¡à 9th = a 8d = 5(a 4d)
a 8d = 5a 20d
a - 5a = 20d - 8d
-4a/-4 = 12d/-4
a = -3d = Relationship.
2a+d=0
To find the relationship between the first term (a) and the common difference (d) in an arithmetic progression (AP), we need to use the given information that the 9th term is five times the 5th term.
Let's denote the 5th term as T5 and the 9th term as T9.
Given: T9 = 5 * T5
In general, the nth term of an AP is given by the formula:
Tn = a + (n-1)d
So, we can rewrite the equation using the formula for the 5th and 9th terms:
a + 8d = 5(a + 4d)
Next, let's simplify the equation by expanding the terms:
a + 8d = 5a + 20d
Now, rearrange the terms:
8d - 20d = 5a - a
Combine like terms:
-12d = 4a
Finally, divide both sides of the equation by 4 to isolate the relationship between a and d:
-12d/4 = 4a/4
-3d = a
Therefore, the relationship between the first term (a) and the common difference (d) in the arithmetic progression is a = -3d.
To find the relationship between "a" (the first term) and "d" (the common difference) in an arithmetic progression (A.P.) given that the 9th term is five times the 5th term, we can follow these steps:
Step 1: Understand the formula for the nth term of an A.P.
The nth term of an A.P. is given by the formula:
Tn = a + (n - 1)d
Here, Tn represents the nth term, a represents the first term, n represents the position of the term in the sequence, and d represents the common difference.
Step 2: Establish the given information.
We are given that the 9th term (T9) is five times the 5th term (T5).
T9 = 5 * T5
Using the formula from Step 1, we can write this equation in terms of "a" and "d" as:
(a + 8d) = 5 * (a + 4d)
Step 3: Simplify and solve the equation.
Distribute the multiplication:
a + 8d = 5a + 20d
Rearrange the terms:
20d - 8d = 5a - a
Simplify:
12d = 4a
Finally, divide both sides of the equation by 4:
3d = a
Therefore, the relationship between "a" and "d" is that "a" is equal to 3 times "d."