The 9th term of an arithmetic progression is three times the 5th term, find the relationship between the first term and common difference
9th term = a+8d
5th term = a + 4d
given: 9th term = 3(5th term)
a + 8d = 3(a + 4d)
a + 8d = 3a + 12d
-2a = 4d
a = -2d, where a is the first term
Do we need to solve for difference
(d)
To find the relationship between the first term and common difference of an arithmetic progression, we need to use the formula for the nth term of an arithmetic progression.
The formula for the nth term of an arithmetic progression is given by:
\[ a_n = a_1 + (n-1) \cdot d \]
where \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.
From the problem statement, we know that the 9th term of the arithmetic progression is three times the 5th term. So we can write this as an equation:
\[ a_9 = 3 \cdot a_5 \]
Substituting the values into the formula for the 9th term and the 5th term, we have:
\[ a_1 + (9-1) \cdot d = 3 \cdot (a_1 + (5-1) \cdot d) \]
Simplifying this equation gives:
\[ a_1 + 8d = 3(a_1 + 4d) \]
Expanding the terms on the right-hand side, we have:
\[ a_1 + 8d = 3a_1 + 12d \]
Combining like terms, we get:
\[ 2a_1 = 4d \]
Dividing both sides of the equation by 2, we find:
\[ a_1 = 2d \]
Therefore, the relationship between the first term (\( a_1 \)) and the common difference (\( d \)) is that the first term is twice the common difference.
To find the relationship between the first term and the common difference in an arithmetic progression, we can use the formula for the nth term of an arithmetic progression.
The formula for the nth term of an arithmetic progression is given by:
tn = a + (n - 1)d
Where tn represents the nth term, a represents the first term, n represents the position of the term, and d represents the common difference.
In this problem, we are given that the 9th term of the arithmetic progression is three times the 5th term. Let's represent the 5th term as t5 and the 9th term as t9.
Given: t9 = 3t5
Using the formula for the nth term, we can write:
t9 = a + (9 - 1)d
t5 = a + (5 - 1)d
Now, substitute the values of t9 and t5 into these equations:
3t5 = a + (9 - 1)d
t5 = a + (5 - 1)d
We can divide the first equation by the second equation to eliminate the term 'a', as follows:
(3t5) / t5 = (a + (9 - 1)d) / (a + (5 - 1)d)
Simplifying this equation, we get:
3 = (a + 8d) / (a + 4d)
Next, cross multiply to eliminate the fraction:
3(a + 4d) = a + 8d
Expand and simplify the equation further:
3a + 12d = a + 8d
Subtract 'a' and '8d' from both sides:
3a - a + 12d - 8d = 0
Simplifying, we get:
2a + 4d = 0
The relationship between the first term (a) and the common difference (d) in the arithmetic progression is 2a + 4d = 0.
Therefore, the relationship can be expressed as 2a = -4d or a = -2d.