the 16th term of an AP is three times the 5th term. If the 12th term is 20 more than the 7th term find the first term and the common difference
Just translate the English into Math, using your definitions and properties of an AP
"the 16th term of an AP is three times the 5th term" ----> a + 15d = 3(a+4d_
a + 15d = 3a + 12d
3d = 2a
a = 3d/2
"the 12th term is 20 more than the 7th term " ---> a + 11d - (a+6d) = 20
5d = 20
....
I will let you finish it ....
Let's denote the first term of the arithmetic progression (AP) as 'a', and the common difference as 'd'.
We are given that the 16th term is three times the 5th term:
a + 15d = 3(a + 4d) (since the 5th term is a + 4d)
Expanding the equation:
a + 15d = 3a + 12d
Simplifying:
15d - 12d = 3a - a
3d = 2a
We are also given that the 12th term is 20 more than the 7th term:
a + 11d = a + 6d + 20 (since the 7th term is a + 6d)
Simplifying:
11d = 6d + 20
5d = 20
d = 4
Plugging the value of 'd' back into the equation 3d = 2a:
3(4) = 2a
12 = 2a
a = 6
Therefore, the first term (a) of the AP is 6, and the common difference (d) is 4.
To find the first term and the common difference in an arithmetic progression (AP), we need to use the given information and the formulas associated with APs.
Let's denote the first term of the AP as 'a' and the common difference as 'd'.
Given information:
1) The 16th term of the AP is three times the 5th term:
The formula to find the nth term of an AP is: Tn = a + (n - 1) * d
According to the given information, we can write the equation as:
a + (16 - 1) * d = 3 * [a + (5 - 1) * d]
Simplifying this equation gives us: a + 15d = 3a + 12d
2) The 12th term is 20 more than the 7th term:
Similarly, using the formula for the nth term, we can write:
a + (12 - 1) * d = a + (7 - 1) * d + 20
Simplifying this equation gives us: a + 11d = a + 6d + 20
To solve the system of equations created by these two equations, we can subtract the second equation from the first equation:
(a + 15d) - (a + 11d) = (3a + 12d) - (a + 6d + 20)
This simplifies to: 4d = 2a - 20
Now, let's solve for the common difference (d) and the first term (a):
From equation (1), we have: a + 15d = 3a + 12d
Rearranging terms gives us: 15d - 12d = 3a - a
This simplifies to: 3d = 2a
Substituting this value in the equation 4d = 2a - 20:
4d = 3d - 20
Further simplifying gives us: d = -20
Now, substituting d = -20 into the equation 3d = 2a:
3(-20) = 2a
-60 = 2a
Dividing both sides by 2, we get: a = -30
Therefore, the first term (a) of the AP is -30, and the common difference (d) is -20.