an Arithmetic progression is given by k,2k/3,k/3,o.find the sixth term,the nth term and if the 20th term is 15 find k
the common difference is -k/3
so, Tn = k-(n-1)(k/3) = k - n/3 k + k/3 = (4-n)k/3
Now find k since T20 = 15
show the steps to follow to get the answers
To find the next term in an arithmetic progression, we need to identify the common difference between each term. The common difference (d) in an arithmetic progression is the constant value added or subtracted to get from one term to the next.
In this case, an arithmetic progression is given by:
k, 2k/3, k/3, o.
To find the common difference (d), we subtract the second term from the first term:
d = (2k/3) - k = 2k/3 - 3k/3 = -k/3
Now, we know that the common difference is -k/3.
1. The sixth term:
To find the sixth term, we add the common difference to the fifth term:
Sixth term = Fifth term + Common difference
The fifth term is the value at the fourth position (since the first term corresponds to the zeroth position in the sequence):
Fifth term = k/3
Common difference = -k/3
Therefore, the sixth term is: (k/3) + (-k/3) = 0.
2. The nth term:
For the nth term, the formula is given by:
nth term = First term + (n - 1) * Common difference
In this case, the first term is k and the common difference is -k/3.
Hence, the nth term is: k + (n - 1) * (-k/3) = k - (n - 1) * k / 3 = (3k - k(n - 1)) / 3.
3. If the 20th term is 15:
To find the value of k when the 20th term is 15, we substitute the values into the nth term formula and solve for k.
15 = (3k - k(20 - 1)) / 3
Simplifying further:
15 = (3k - 19k) / 3
15 = -16k / 3
Now, multiply both sides of the equation by 3:
45 = -16k
Finally, divide both sides by -16 to solve for k:
k = 45 / -16
Therefore, the value of k is -45/16.