An A . P is given by k, 2k/3,k/3,0... r
(I)find the sixth term
(Ii)if the 20 term is equal to 15,find k.
To find the sixth term of the AP, we need to find the pattern of the AP.
The given pattern is k, 2k/3, k/3, 0... r.
From this pattern, we can see that the common difference between each term is -k/3.
Therefore, to find the sixth term, we can add (-k/3) to the third term:
6th term = k/3 + (-k/3)
= 0
So, the sixth term is 0.
To find the value of k when the 20th term is equal to 15, we can use the formula for the nth term of an AP:
n-th term = a + (n-1)d
where a is the first term, d is the common difference, and n is the term number.
From the given pattern, we know that the first term is k and the common difference is -k/3.
Using the formula, we can substitute the values:
20th term = k + (20-1)(-k/3)
= k - (19/3)k
= (1 - 19/3)k
= (3/3 - 19/3)k
= (-16/3)k
Given that the 20th term is equal to 15, we can set up the equation:
(-16/3)k = 15
To solve for k, we can multiply both sides of the equation by -3/16:
k = 15 * (-3/16)
= -45/16
Therefore, k is equal to -45/16.
The first term of geometric series is 4.what is the sum of the first 6 term
of the series if the common ratio is 3 .
To find the sum of the first 6 terms of a geometric series, we can use the formula:
S_n = a * (r^n - 1) / (r - 1)
Where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
In this case, the first term (a) is 4 and the common ratio (r) is 3. We want to find the sum of the first 6 terms (n = 6).
Plugging these values into the formula, we get:
S_6 = 4 * (3^6 - 1) / (3 - 1)
Simplifying this equation, we have:
S_6 = 4 * (729 - 1) / 2
= 4 * 728 / 2
= 1456
So, the sum of the first 6 terms of the geometric series is 1456.
To find the sixth term of the arithmetic progression (AP) defined by k, 2k/3, k/3, 0, ..., r, we can use the formula for the nth term of an AP:
Tn = a + (n - 1)d
where Tn is the nth term of the AP, a is the first term, n is the position of the term in the AP, and d is the common difference.
For this AP, the first term a is k, and the common difference d is k/3 - 0 = k/3.
(I) To find the sixth term, we substitute a = k, n = 6, and d = k/3 into the formula:
T6 = k + (6 - 1)(k/3)
= k + 5k/3
= (3k + 5k)/3
= 8k/3
Therefore, the sixth term of the AP is 8k/3.
(II) If the 20th term is equal to 15, we can use the same formula to find k. Now, we substitute Tn = 15, a = k, n = 20, and d = k/3 into the formula:
15 = k + (20 - 1)(k/3)
= k + 19k/3
= (3k + 19k)/3
= 22k/3
To solve for k, we can now solve the equation 22k/3 = 15:
22k = 3 * 15
k = (3 * 15) / 22
k ≈ 2.045
Therefore, if the 20th term is equal to 15, k is approximately 2.045.