the 6th term of A.P. is 5 times the fifth term and the 11th term is exceeds twice the fifth term by three.find the 8th term
6th term: 16,-4, 1, -1/4
To solve this problem, let's first find the common difference (d) of the arithmetic progression (A.P.).
Let the fifth term be a.
According to the given information, the sixth term is 5 times the fifth term: 6th term = 5a.
Also, the 11th term exceeds twice the fifth term by three: 11th term = 2a + 3.
We can write the formula for the nth term of an arithmetic progression as: aₙ = a + (n - 1)d.
So, we have two equations:
Equation 1: 6th term = 5a ( 6th term = a + 5d)
Equation 2: 11th term = 2a + 3 ( 11th term = a + 10d)
Now, we can solve these equations simultaneously to find the values of a and d.
From Equation 1:
a + 5d = 5a
Rearranging the terms:
4a = 5d
From Equation 2:
a + 10d = 2a + 3
Rearranging the terms:
a = 3 - 8d
Substituting this value of a into the equation 4a = 5d:
4(3 - 8d) = 5d
Expanding and simplifying:
12 - 32d = 5d
37d = 12
d ≈ 0.324
Now, substitute the value of d back into either of the equations to find the value of a.
From Equation 1:
a + 5(0.324) = 5a
Simplifying:
a = 1.62
We've found that a ≈ 1.62 and d ≈ 0.324.
Now let's find the 8th term (8th term = a + 7d):
8th term = 1.62 + 7(0.324)
8th term ≈ 1.62 + 2.268
8th term ≈ 3.888
Therefore, the 8th term of the arithmetic progression is approximately 3.888.
To find the 8th term of the arithmetic progression (AP), we need to first determine the common difference (d) of the sequence.
Let's denote the first term of the AP as "a" and the common difference as "d". Since we are given that the 6th term is 5 times the 5th term, we can write the following equation:
a + 5d = 5(a + 4d)
Next, we are given that the 11th term exceeds twice the 5th term by three:
a + 10d = 2(a + 4d) + 3
Now, we have a system of two equations. We can solve this system to find the values of "a" and "d".
Solving the first equation (a + 5d = 5a + 20d) gives us:
4a = 15d (Equation 1)
Solving the second equation (a + 10d = 2a + 8d + 3) gives us:
a = 2d + 3 (Equation 2)
Now, we can substitute Equation 2 into Equation 1:
4(2d + 3) = 15d
8d + 12 = 15d
12 = 7d
d = 12 / 7
d ≈ 1.714
Now that we have the value of "d", we can find the value of "a" by substituting it back into Equation 2:
a = 2(1.714) + 3
a ≈ 5.429
Therefore, the first term (a) is approximately 5.429 and the common difference (d) is approximately 1.714.
Now, we can find the 8th term by using the formula for the nth term of an AP:
an = a + (n - 1)d
Substituting the given values:
a8 = 5.429 + (8 - 1) * 1.714
a8 ≈ 5.429 + 6 * 1.714
a8 ≈ 5.429 + 10.284
a8 ≈ 15.713
Therefore, the 8th term of the AP is approximately 15.713.
a+5d = 5(a+4d)
a+10d = 2(a+4d)+3
Now just solve for a and d, and figure
T8 = a+7d