Find a nth term for this sequence 1/3,1/9,1/15,1/21,1/27..
How
To find the nth term of this sequence, we need to look for a pattern. In this case, it seems that each term is the reciprocal of a number that follows a certain pattern. If we observe the denominator of each term, we can notice that it is increasing by 6 each time.
Let's take a closer look:
1/3 = 1/(3*1)
1/9 = 1/(3*3)
1/15 = 1/(3*5)
1/21 = 1/(3*7)
1/27 = 1/(3*9)
The denominator of each term is the product of 3 and an odd number (1, 3, 5, 7, 9), which starts from 1 and increases by 2 each time.
Therefore, we can express the nth term as 1 / (3 * ((2 * n) - 1)).
So, the general formula for the nth term of this sequence is 1 / (3 * ((2 * n) - 1)).
To find the nth term in this sequence, we need to find a pattern in the numbers.
In this sequence, each term is obtained by taking the reciprocal of an arithmetic sequence starting from 3 and increasing by 6 each time.
Let's break it down step-by-step:
1st term: The arithmetic sequence starts from 3, so the 1st term is 3.
2nd term: The arithmetic sequence increases by 6, so the 2nd term is 3 + 6 = 9.
3rd term: The arithmetic sequence increases by 6, so the 3rd term is 9 + 6 = 15.
4th term: The arithmetic sequence increases by 6, so the 4th term is 15 + 6 = 21.
5th term: The arithmetic sequence increases by 6, so the 5th term is 21 + 6 = 27.
In general, the nth term of the arithmetic sequence can be found using the formula: tn = a + (n-1)d,
where tn is the nth term, a is the first term, n is the position of the term, and d is the common difference.
In this case, a = 3 and d = 6.
So, the nth term of the arithmetic sequence is: tn = 3 + (n-1)(6) = 3 + 6n - 6 = 6n - 3.
Finally, to find the nth term of the given sequence, we take the reciprocal of the nth term of the arithmetic sequence:
nth term = 1/(6n - 3).
Therefore, the nth term for the given sequence 1/3, 1/9, 1/15, 1/21, 1/27.. is 1/(6n - 3).