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)
Not enough workings😟
I still don't get you
You are madoo
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.