the 3rd term of a gp is 63 and 5th term is 567.find the sum of the 6th term of the progression
r^2 = 567/63 = 9
r = ±3
Now you can find a, and
S6 = a(r^6-1)/(r-1)
Full answer
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Well, well, well, looks like we've got ourselves a geometric progression (GP). Let's get cracking and find the common ratio, shall we?
To find the common ratio, we'll divide the 5th term (567) by the 3rd term (63). Now let me grab my trusty calculator:
567 divided by 63 equals... drumroll, please... 9! So the common ratio is 9.
To find the 6th term, we can multiply the 5th term (567) by the common ratio (9). Let's do some quick math:
567 multiplied by 9 equals... *drum roll intensifies* 5103! Voila, we have the 6th term.
Now, you wanted to find the sum of the 6th term, so let's throw in some more math wizardry. The sum of the terms of a GP can be found using the formula:
Sum = a * ((r^n) - 1) / (r - 1)
where a is the first term, r is the common ratio, and n is the number of terms. In this case, we're looking for the sum of the 6th term, so n = 6.
Now, my dear friend, what's the first term (a) of this GP? You didn't mention it.
To find the sum of the 6th term of a geometric progression (GP), we need to know the first term and the common ratio. However, these values have not been provided.
To find the sum of the 6th term, we must first find the first term and the common ratio using the given information. Here's how:
Given that the 3rd term of the GP is 63 and the 5th term is 567, we can use the formula for the nth term of a GP:
an = a1 * r^(n-1)
where 'an' is the nth term, 'a1' is the first term, 'r' is the common ratio, and 'n' is the position of the term.
Using this formula, we can create two equations using the given information:
63 = a1 * r^2 (Equation 1)
567 = a1 * r^4 (Equation 2)
To solve these equations simultaneously, we need to eliminate 'a1'. Divide Equation 2 by Equation 1:
(567/63) = (a1 * r^4) / (a1 * r^2)
Simplifying:
9 = r^2
Taking the square root of both sides:
r = ±3
Now, substitute the value of 'r' back into either Equation 1 or Equation 2 to find 'a1'.
Using Equation 1:
63 = a1 * (±3)^2
63 = 9a1
Dividing both sides by 9:
a1 = 7
Now that we have found the values of 'a1' and 'r' as 7 and 3, respectively, we can find the sum of the 6th term using the formula for the sum of the first 'n' terms of a GP:
Sn = a1 * (1 - r^n) / (1 - r)
Substituting the values:
S6 = 7 * (1 - 3^6) / (1 - 3)
Simplifying:
S6 = 7 * (1 - 729) / (1 - 3)
S6 = 7 * (-728) / (-2)
S6 = 7 * 364 = 2548
Therefore, the sum of the 6th term of the geometric progression is 2548.