Find the 7th term of harmonic sequence whose 2nd term is 2 and 5th term is - 1/2.
so, the corresponding AP is
? 1/2 ? ? -2
3d = -5/2, so d = -5/6 and the AP is
4/3, 1/2, -1/3, -7/6, -2, -17/6, -11/3
so the 7th term of the HS is -3/11
In harmonic sequence:
a1 = 1 / a
a2 = 1 / ( a + d )
a3 = 1 / ( a + 2 d )
a4 = 1 / ( a + 3 d )
a5 = 1 / ( a + 4 d )
a6 = 1 / ( a + 5 d )
a7 = 1 / ( a + 6 d )
in this case:
a2 = 1 / ( a + d ) = 2
a5 = 1 / ( a + 4 d ) = - 1 / 2
Second term:
1 / ( a + d ) = 2
Multiply both sides by ( a + d )
1 = 2 ( a + d )
1 = 2 a + 2 d
2 a + 2 d = 1
Fifth term:
1 / ( a + 4 d ) = - 1 / 2
Multiply both sides by ( a + 4 d )
1 = - ( a + 4 d ) / 2
Multiply both sides ba 2
2 = - ( a + 4 d )
Multiply both sides by - 1
- 2 = a + 4 d
a + 4 d = - 2
Now you must solve system:
2 a + 2 d = 1
a + 4 d = - 2
__________
The solution is:
a = 4 / 3 , d = - 5 / 6
a7 = 1 / ( a + 6 d ) = 1 / [ 4 / 3 + 6 ∙ ( - 5 / 6 ) ] = 1 / ( 4 / 3 - 5 ) =
1 / ( 4 / 3 - 15 / 3 ) = 1 / ( - 11 / 3 ) = - 3 / 11
To find the 7th term of a harmonic sequence, we need to determine the common difference between consecutive terms.
Let's start by finding the common difference using the given information.
Given:
2nd term = 2
5th term = -1/2
The formula for the nth term of a harmonic sequence is:
Tn = 1 / (a + (n-1)d)
where:
Tn = nth term
a = 1st term
d = common difference
Using the formula, we can set up two equations using the given information:
T2 = 2
T5 = -1/2
2 = 1 / (a + (2-1)d) -- (equation 1)
-1/2 = 1 / (a + (5-1)d) -- (equation 2)
Now, let's solve these equations to find the values of "a" and "d".
From equation 1:
2 = 1 / (a + d)
From equation 2:
-1/2 = 1 / (a + 4d)
Now, we can cross multiply and solve these equations:
Equation 1:
2(a + d) = 1
2a + 2d = 1
Equation 2:
-1/2(a + 4d) = 1
-a - 2d = 2
Now, let's solve these equations simultaneously.
Multiplying Equation 1 by 2:
4a + 4d = 2
Adding Equation 2 to the new Equation 1:
4a + 4d + (-a) - 2d = 2 + 2
3a + 2d = 4
Now, we have a system of linear equations:
2a + 2d = 1 -- (equation 1)
3a + 2d = 4 -- (equation 2)
Using the method of substitution or elimination, we can solve these equations to find the values of "a" and "d".
Let's use the method of elimination:
Multiply equation 1 by -3:
-6a - 6d = -3
Adding equation 2 and the new equation 1:
-6a - 6d + 3a + 2d = -3 + 4
-3a - 4d = 1
Rearranging the equation:
3a + 4d = -1 -- (equation 3)
Now, we have two equations:
3a + 4d = -1 -- (equation 3)
3a + 2d = 4 -- (equation 2)
Subtracting equation 3 from equation 2:
(3a + 2d) - (3a + 4d) = 4 - (-1)
3a + 2d - 3a - 4d = 4 + 1
-2d = 5
d = -5/2
Now, let's substitute the value of d into equation 1 to find the value of a:
2a + 2(-5/2) = 1
2a - 5 = 1
2a = 1 + 5
2a = 6
a = 6/2
a = 3
So, the first term, a = 3, and the common difference, d = -5/2.
Now, let's find the 7th term of the sequence using the formula:
Tn = 1 / (a + (n-1)d)
T7 = 1 / (3 + (7-1)(-5/2))
T7 = 1 / (3 + 6(-5/2))
T7 = 1 / (3 - 15/2)
T7 = 1 / (6/2 - 15/2)
T7 = 1 / (-9/2)
T7 = -2/9
Therefore, the 7th term of the given harmonic sequence is -2/9.
To find the 7th term of a harmonic sequence, we first need to find the common difference (d) of the sequence.
In a harmonic sequence, the nth term (a_n) is given by the formula: a_n = 1 / (a + (n - 1)d), where a is the first term and d is the common difference.
Given that the 2nd term (a_2) is 2 and the 5th term (a_5) is -1/2, we can set up the following equations:
For the 2nd term (a_2):
a_2 = 1 / (a + d) = 2
For the 5th term (a_5):
a_5 = 1 / (a + 4d) = -1/2
We can solve these two equations simultaneously to find the values of a and d.
First, let's solve for a in terms of d from the equation for the 2nd term:
1 / (a + d) = 2
Multiplying both sides by (a + d), we get:
1 = 2(a + d)
1 = 2a + 2d
2a + 2d = 1
a + d = 1/2 (Equation 1)
Now, let's solve for a in terms of d from the equation for the 5th term:
1 / (a + 4d) = -1/2
Multiplying both sides by (a + 4d), we get:
1 = -1/2(a + 4d)
1 = (-1/2)a - 2d
(-1/2)a - 2d = 1
a + 4d = -2 (Equation 2)
Now we have a system of equations (Equation 1 and Equation 2) that we can solve to find the values of a and d.
Multiplying Equation 1 by 2, we get:
2a + 2d = 1
Multiplying Equation 2 by -2, we get:
-2a - 8d = 4
Adding these two equations, we eliminate the variable a:
2a - 2a + 2d - 8d = 1 + 4
-6d = 5
d = -5/6
Substituting the value of d back into Equation 1:
a + d = 1/2
a - 5/6 = 1/2
a = 1/2 + 5/6
a = 3/2
Now that we have found the values of a and d, we can find the 7th term (a_7) using the formula for the nth term (a_n) in a harmonic sequence:
a_n = 1 / (a + (n - 1)d)
Substituting the values of a, d, and n = 7 into the formula:
a_7 = 1 / (3/2 + (7 - 1)(-5/6))
Simplifying:
a_7 = 1 / (3/2 - 6(-5/6))
a_7 = 1 / (3/2 + 5)
Invert and multiply:
a_7 = 1 * (2/3 + 5/2)
a_7 = 2/3 + 5/2
a_7 = (2 * 2 + 5 * 3) / (3 * 2)
a_7 = (4 + 15) / 6
a_7 = 19/6
Therefore, the 7th term of the given harmonic sequence is 19/6.