if a and d are the first term and the common difference of an arithmetic sequence, respectively, what is the nth term of the corresponding harmonic sequence?
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if a and d are the first term and the common difference of an arithmetic sequence, respectively, what is the nth term of
Answer and Step-by-step explanation:
To find the nth term of the given harmonic sequence, convert it first into an arithmetic sequence by taking their reciprocals.
Harmonic sequence:
1, 2/3, 1/2, 2/5, ..., n = 10
Arithmetic sequence:
1, 3/2, 2, 5/2, ..., n = 10
Then find the common difference by subtracting a term by the previous term.
3/2 - 1 = 1/2
2 - 3/2 = 1/2
5/2 - 2 = 1/2
Thus, the common difference is 1/2.
Then use the arithmetic sequence formula to find the nth term:
An = A1 + (n - 1)d
where
An is the nth term,
A1 is the first term,
n is the number of terms and
d is the common difference.
Substitute the given values to the formula. Then simplify.
An = An
A1 = 1
n = n
d = 1/2
An = 1 + (n - 1)(1/2)
An = 1 + 1/2n - 1/2
An = 1/2 + 1/2n
The nth term of the given harmonic sequence is An = 1/2 + 1/2n. Make sure to take the reciprocal after using this formula.