The sum of the first 8 terms of the A.P logx,logx²,logx³,.......is
log(x^n) = n log(x)
so the A.P. is ... log(x), 2 log(x), 3 log(x), ....
8th term is ... 8 log(x)
sum is ... 36 log(x)
I still don't get you
You are madoo
Not enough workings😟
Why did the logarithm go to the circus?
Because it wanted to join the arithmetic progression party!
But enough clowning around, let's calculate the sum of the first 8 terms of the arithmetic progression. The formula for the sum of an arithmetic progression is given by S = (n/2)(2a + (n-1)d), where S is the sum, n is the number of terms, a is the first term, and d is the common difference.
In this case, a = logx, n = 8, and d = logx² - logx. Now let's plug these values into the formula and see what we get.
S = (8/2)(2logx + (8-1)(logx² - logx))
Simplifying further:
S = 4(2logx + (7logx² - 7logx))
S = 8logx + 28logx² - 28logx
So, the sum of the first 8 terms of the given arithmetic progression is 8logx + 28logx² - 28logx.
Now that we have the answer, I hope it didn't make your head spin like a circus wheel!
To find the sum of the first 8 terms of an arithmetic progression (A.P.) logx, logx², logx³,..., we need to determine the general formula for the nth term, and then use the formula for the sum of an arithmetic series.
The given A.P. has a common ratio of r = 2, as each term is the square of the previous term (e.g., logx² = (logx)²).
Now, let's find the general formula for the nth term of the A.P. logx, logx², logx³,...:
The first term is logx (when n = 1), and the common ratio is r = 2.
Using the formula for the nth term of an A.P., 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 (which in this case is logx), we have:
aₙ = logx + (n-1)logx
aₙ = n*logx
Now we can find the sum of the first 8 terms using the formula for the sum of an arithmetic series:
Sₙ = (n/2)(a₁ + aₙ), where Sₙ is the sum of the first n terms.
Substituting the values, we have:
S₈ = (8/2)(logx + 8*logx)
S₈ = 4(logx + 8logx)
S₈ = 4(9logx) {combining like terms}
S₈ = 36logx {simplifying further}
Therefore, the sum of the first 8 terms of the given A.P. is 36logx.