If 6 times the sixth term of an arithmetic progression is equal to 9 times the 9th term, find
the 15th term.
6(a+5d)=9(a+8d)
6a+30d=9a+72d
3a = -42d
a = -14d
So, there are lots of answers. For example, if d=1, a=-14 and the sequence is
-14,-13-,12,-11,-10,-9,-8,-7,-6, ...
6(-9)=9(-6)
To find the 15th term of the arithmetic progression, we need to find the common difference of the sequence first.
Let's assume that the first term of the arithmetic progression is 'a' and the common difference is 'd'.
Therefore,
Sixth term = a + 5d (as the sixth term is the sum of the first term and 5 times the common difference)
Ninth term = a + 8d (as the ninth term is the sum of the first term and 8 times the common difference)
According to the given condition: 6(a + 5d) = 9(a + 8d)
Simplifying the equation, we get:
6a + 30d = 9a + 72d
Rearranging the terms:
3a = 42d
a = 14d/3
Now that we have the value of 'a' in terms of 'd', we can use it to find the 15th term:
Fifteenth term = a + 14d (as the fifteenth term is the sum of the first term and 14 times the common difference)
Substituting the value of 'a' into the equation:
Fifteenth term = (14d/3) + 14d
Fifteenth term = (14d + 42d)/3
Fifteenth term = 56d/3
Therefore, the 15th term of the arithmetic progression is 56d/3.
To find the 15th term of an arithmetic progression, we need to determine the first term and the common difference.
Let's start by using the given information that "6 times the sixth term is equal to 9 times the 9th term." We can express this in mathematical terms as follows:
6 * a + 5 * d = 9 * a + 8 * d
Here, a represents the first term of the arithmetic progression, and d represents the common difference. We have used the formula that the nth term of an arithmetic progression can be expressed as a + (n - 1) * d.
Now we need another piece of information to solve for a and d. The given problem doesn't provide any further information directly, so we need to use a system of equations.
One possible approach is to create another equation using the fact that the 15th term (a15) is equal to a + 14 * d (since it is the first term plus 14 times the common difference):
a + 14 * d = ?
To solve for a and d, we can substitute the value of a in the first equation into the second equation. Rearranging the first equation, we get:
6a - 9a = 8d - 5d
Simplifying,
-3a = 3d
Now, substitute this value of a into the second equation:
3d + 14d = ?
Simplifying,
17d = ?
Since we don't have a specific value for d, we can't find the exact value of the 15th term. However, we can express it in terms of d as:
a15 = a + 14 * d = 0 + 14d = 14d
So, the 15th term of the arithmetic progression is 14 times the common difference (14d).