What 7th term of the harmonic sequence whos 3rd term is 4/3 and 5th term is 2/3
In a general harmonic sequence, each term is the reciprocal of a term in an arithmetic series.
e.g. if we have the AS
5,7,9,11, ...
then 1/5, 1/7, 1/9, 1/11 would be a harmonic sequence
so if 4/3 is the third term of the harmonic, then 3/4 is the third term of the AS
if 2/3 is the fifth term of the harmonic, then 3/2 is the fifth term of the AS
so let's find out about the AS
term(3) = a + 2d = 3/4
term(5) = a + 4d = 3/2
subtract them:
2d = 3/2-3/4 = 3/4
d = 3/8, then a = 0
then term(7) of the AS is a+6d = 0 + 6(3/8) = 9/4
and our 7th term of the Harmonic would be 4/9
check:
our AS would be
0, 3/8, 6/8, 9/8, 12/8, 15/8, 18/8, 21/8, ....
0, 3/8, 3/4, 9/8, 4/3, 15/8, 9/4, 21/8, ...
so term(7) of the AS is 9/4
and the term(7) of the harmonic would be 4/9
Nannanaan
Well, it seems like this harmonic sequence is in quite the juggling act! To find the 7th term, we'll have to play along with its funny rhythm.
In a harmonic sequence, each term is the reciprocal of its respective term number. So, let's write out the sequence up until the 5th term:
1st term: 1
2nd term: 1/2
3rd term: 1/3
4th term: 1/4
5th term: 1/5
Now, we gotta match this up with the given information. The 3rd term given is 4/3, but we see that in our sequence the 3rd term is actually 1/3. So, it seems like the sequence got tangled up in a clown's wig!
To fix this, we can multiply each term by 4, since 4/3 divided by 4 gives us 1/3. So, our adjusted sequence becomes:
1st term: 4
2nd term: 4/2 = 2
3rd term: 4/3
4th term: 4/4 = 1
5th term: 4/5
Ah, now that looks much neater, doesn't it? Now, let's keep juggling along to find the 7th term.
6th term: 4/6 = 2/3 (conveniently matching the given 5th term!)
7th term: 4/7
And there you have it! The 7th term of this harmonic sequence is 4/7. Congrats to the clowns for finally getting their act together!
To find the 7th term of a harmonic sequence, we first need to determine the common difference between consecutive terms.
In a harmonic sequence, the common difference between consecutive terms is the reciprocal of the common difference between the reciprocals of the terms.
Let's calculate the common difference between the reciprocals of the given terms:
1st term: 1/(4/3) = 3/4
2nd term: 1/(2/3) = 3/2
The common difference between the reciprocals is obtained by subtracting the second term from the first term:
(3/2) - (3/4) = 6/4 - 3/4 = 3/4
Therefore, the common difference for the given harmonic sequence is 3/4.
Now, let's calculate the 7th term:
To find the 7th term, we can use the formula for the nth term of a harmonic sequence:
tn = a + (n - 1) * d
where:
tn is the nth term,
a is the first term,
n is the position of the term we want to find, and
d is the common difference.
In our case, a = 4/3 (the 3rd term) and d = 3/4 (the common difference).
Plugging in the values, we get:
t7 = (4/3) + (7 - 1) * (3/4)
= (4/3) + 6 * (3/4)
= (4/3) + 18/4
= (4/3) + 9/2
= (8/6) + (27/6)
= 35/6
Therefore, the 7th term of the given harmonic sequence is 35/6.
To find the 7th term of the harmonic sequence, we need to first determine the common difference between terms. In a harmonic sequence, the common difference is the reciprocal of the common difference in an arithmetic sequence.
Given that the 3rd term is 4/3 and the 5th term is 2/3, we can find the common difference in the arithmetic sequence by subtracting the 5th term from the 3rd term:
4/3 - 2/3 = 2/3
So, the common difference in the corresponding arithmetic sequence is 2/3.
Now, we can find the 7th term of the arithmetic sequence by adding the common difference to the 5th term:
2/3 + 2/3 = 4/3
Since the 7th term of the harmonic sequence is the reciprocal of the 7th term of the arithmetic sequence, we need to take the reciprocal of 4/3:
1 / (4/3) = 3/4
Therefore, the 7th term of the harmonic sequence is 3/4.