Question

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.

Answer 1

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 .

Answer 2

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.

Answer 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.