The sum of the 1st eight terms of an arithmetic progression is logx, logxsquare, logxcube is find the sum of the 1st 8 term of an arithmetic progression
To find the sum of the first 8 terms of an arithmetic progression, we need to determine the first term (a) and the common difference (d).
Let's analyze the given information:
1st term: log(x)
2nd term: log(x^2)
3rd term: log(x^3)
Observing the pattern, we can see that each term is the logarithm of x raised to a power equal to its position in the progression. This suggests that the common ratio (r) between consecutive terms is x.
Since the first term is log(x), we can write it as a = log(x).
Using the formula for the nth term of an arithmetic progression, we have:
a_n = a + (n-1)d
We can plug in the values for the 2nd and 3rd terms to find the common difference (d):
log(x^2) = log(x) + (2-1)d
log(x^3) = log(x) + (3-1)d
Simplifying these equations, we have:
log(x^2) = log(x) + d
log(x^3) = log(x) + 2d
We can subtract log(x) from both equations to eliminate it:
log(x^2) - log(x) = d
log(x^3) - log(x) = 2d
Using the logarithmic property log(a^b) = b * log(a):
log(x^2/x) = d
log(x^3/x) = 2d
Simplifying further:
log(x) = d
log(x^2) = 2d
Now that we know the common difference is d = log(x), we can find the sum of the first 8 terms using the formula. Recall that the sum of an arithmetic progression, Sn, can be calculated using the formula:
Sn = n / 2 * (2a + (n-1)d)
Plugging in the values, we have:
a = log(x)
d = log(x)
S8 = 8 / 2 * (2log(x) + (8-1)log(x))
= 8 / 2 * (2log(x) + 7log(x))
= 8 / 2 * (9log(x))
= 4 * 9log(x)
= 36log(x)
Therefore, the sum of the first 8 terms of the arithmetic progression is 36log(x).
To find the sum of the first eight terms of an arithmetic progression, we need to know the first term (a) and the common difference (d). However, the given information states the sum of the terms in logarithmic form.
To convert logarithmic form to arithmetic form, we need to use the properties of logarithms. We know that log(x^n) = n * log(x), where log is the logarithm to the base 10.
Using this property, we can rewrite the given terms in arithmetic form:
log(x) = log(x^1)
log(x^2) = 2 * log(x)
log(x^3) = 3 * log(x)
Now, we can identify the common ratio by comparing the exponents of x in the logarithmic terms:
Common ratio = (2 * log(x)) / log(x) = 2
Since the common ratio is 2, we can say that the common difference (d) is equal to (2 - 1) = 1.
Now, we have the first term (a) as log(x) and the common difference (d) as 1. To find the sum of the first eight terms, we can use the formula for the sum of an arithmetic progression:
Sum = (n/2) * (2a + (n-1)d)
Plugging in the values, we get:
Sum = (8/2) * (2log(x) + (8-1) * 1)
= 4 * (2log(x) + 7)
= 8log(x) + 28
Therefore, the sum of the first eight terms of the arithmetic progression is 8log(x) + 28.