An Ap is given by K,2k/3,K/3,0.....Find the 6th term, Find the nth term. If the 20th term is equal to 15 find K
Oh, you want me to find K? Alright, let's start with finding the 6th term of the arithmetic progression (AP) you provided.
The AP is given by K, 2K/3, K/3, 0.
To find the 6th term, we can use the formula for the nth term of an arithmetic progression: An = A1 + (n - 1)d. Here, A1 is the first term, n is the number of the term we want to find, and d is the common difference between consecutive terms.
In this case, A1 = K, n = 6, and d = -K/3.
So, the 6th term (A6) will be:
A6 = K + (6 - 1)(-K/3)
A6 = K - K/3
A6 = 2K/3
Now, let's find the nth term of this AP. Again, we'll use the formula An = A1 + (n - 1)d.
We want to find the nth term, so the formula becomes:
An = K + (n - 1)(-K/3)
Finally, if the 20th term is equal to 15, we can substitute n = 20 and equate it to 15 to find K:
15 = K + (20 - 1)(-K/3)
15 = K - 19K/3
15 = (3K - 19K)/3
45 = -16K
K = -45/16
So, K is equal to -45/16.
To find the 6th term of the arithmetic progression (AP), we can use the formula for the nth term of an AP:
an = a1 + (n-1)d
where an is the nth term, a1 is the first term, n is the position of the term, and d is the common difference.
Given that the AP is given by K, 2k/3, K/3, 0, ...,
a1 = K
d = (2k/3) - K = K/3
To find the 6th term (a6), substitute these values into the formula:
a6 = K + (6-1)(K/3)
= K + 5K/3
= 8K/3
Therefore, the 6th term of the AP is 8K/3.
To find the nth term of the AP, we can use the same formula:
an = a1 + (n-1)d
Substitute the given values:
an = K + (n-1)(K/3)
If we want to find the value of K when the 20th term (a20) is equal to 15:
a20 = 15
K + (20-1)(K/3) = 15
Simplify the equation:
K + 19K/3 = 15
Multiply through by 3:
3K + 19K = 45
22K = 45
Divide both sides by 22:
K = 45/22
Therefore, the value of K when the 20th term is equal to 15 is K = 45/22.
To find the 6th term of the arithmetic progression (AP), we need to understand the pattern in the given sequence.
The given AP is: K, 2k/3, K/3, 0, ...
We can observe that each term is obtained by subtracting k/3 from the previous term.
Let's list out the first few terms:
1st term = K
2nd term = 2K/3 = (K - K/3)
3rd term = K/3 = (2K/3 - K/3)
4th term = 0 = (K/3 - K/3)
We can notice the pattern:
1st term - 2nd term = 2nd term - 3rd term = 3rd term - 4th term = ...
Therefore, we can conclude that the common difference of the AP is k/3.
To find the 6th term, we start with the first term and add the common difference five times:
6th term = 1st term + (6 - 1) * common difference
= K + 5 * (k/3)
= K + 5k/3
= (3K + 5k) / 3
Hence, the 6th term of the AP is (3K + 5k) / 3.
To find the nth term of an arithmetic progression, we can use the formula:
nth term = 1st term + (n - 1) * common difference
So, the nth term of the given AP is
nth term = K + (n - 1) * (k/3)
= (3K + k(n - 1))/3
Given that the 20th term is 15, we can substitute it into the nth term formula:
15 = (3K + k(20 - 1))/3
15 = (3K + 19k)/3
To eliminate the fraction, we can multiply both sides of the equation by 3:
45 = 3K + 19k
Simplifying further:
45 = 22K + 19k
To find the value of K, we need to solve this equation.
common difference = 2k/3 - k = -k/3
given: 20th term is 15
so k + 19(-k/3) = 15
times 3, (I don't like fractions)
3k - 19k = 45
k = - 45/16
first term is -45/16
common difference = -k/3 = 15/16
term 6 = a + 5d
= -45/16 + 5(15/16) = 15/8
term n = a + (n-1)d = a + nd - d
= -45/12 + (15/16)n - 15/16
= -75/16 + (15/16)n
or
= 15(n - 5)/16 or .... (15/16)(n-5)
take your pick