find the sum of n arithmetic means between x and 3x.

(answer 2nx)

Hazaribag

To find the sum of n arithmetic means between x and 3x, we first need to determine the common difference (d) between consecutive terms.

Since there are n arithmetic means between x and 3x, we can divide the difference between 3x and x by the number of terms, which is n + 1 (including x and 3x):

d = (3x - x) / (n + 1)
d = 2x / (n + 1)

The sum of an arithmetic series can be calculated using the formula:
Sn = (n/2)(2a + (n - 1)d)

In this case, the first term (a) is x, and the common difference (d) is 2x / (n + 1). Substituting these values into the formula:

Sn = (n/2)(2x + (n - 1)(2x / (n + 1)))
Sn = (n/2)(2x + (2nx - 2x) / (n + 1))
Sn = (n/2)(2x + (2nx - 2x) / (n + 1))
Sn = (n/2)((2x(n + 1) + 2nx - 2x) / (n + 1))
Sn = (n/2)((2nx + 2x + 2nx - 2x) / (n + 1))
Sn = (n/2)(4nx / (n + 1))
Sn = 2n(nx / (n + 1))
Sn = 2n(x(n / (n + 1)))

Simplifying further, we have:
Sn = 2n(x(n) / (n + 1))
Sn = 2nx(n / (n + 1))
Sn = 2nx

Therefore, the sum of n arithmetic means between x and 3x is 2nx.