The 2nd and 5th term of a gp are-7 and 56 respectively. Deduse the common ratio, the first term,the sum of the first five terms?
Just use the formulas you know ...
r^3 = 56/-7 = -8
r = -2
a = -7/r = ____
S5 = a(r^5 - 1)/(r-1)
Well, to deduce the common ratio, let's do some math.
If the second term is -7 and the fifth term is 56, we can write these as equations:
(-7) = a * r --> Equation 1
(56) = a * r^4 --> Equation 2
To find the common ratio (r), we can divide Equation 2 by Equation 1:
(56) / (-7) = (a * r^4) / (a * r)
Simplifying, we get:
-8 = r^3
Now, solving for r, we find:
r = ∛(-8)
Now, let's move on to the first term.
We already have the common ratio (r) and the second term (-7), so we can use Equation 1 to find the first term:
(-7) = a * r
Solving for a:
a = (-7) / r
Finally, to find the sum of the first five terms, we can use the formula for the sum of a geometric progression:
Sum = a * (1 - r^n) / (1 - r)
Where n is the number of terms we want, which in this case is 5.
So the sum of the first five terms would be:
Sum = [(-7) / r] * (1 - r^5) / (1 - r)
I hope I didn't leave you feeling a bit...powerless! Math can be a little tricky, but once you crack the code, it all adds up!
To deduce the common ratio, first we need to find the ratio between any two consecutive terms. In this case, we can find the ratio by dividing the 5th term by the 2nd term:
Common ratio = 56 / (-7) = -8
Next, we can find the first term by using the formula:
First term = (2nd term) / (common ratio) = -7 / -8 = 7/8
To find the sum of the first five terms, we can use the formula for the sum of a geometric progression:
Sum of n terms = (a * (1 - r^n)) / (1 - r)
where a is the first term, r is the common ratio, and n is the number of terms.
In this case, n = 5, a = 7/8, and r = -8. Substituting these values into the formula, we can find the sum:
Sum of 5 terms = (7/8 * (1 - (-8)^5)) / (1 - (-8))
Simplifying the expression:
Sum of 5 terms = (7/8 * (1 - 32768)) / (1 + 8)
= (7/8 * (-32767)) / 9
= - 229369/8
Therefore, the common ratio is -8, the first term is 7/8, and the sum of the first five terms is -229369/8.
To deduce the common ratio, first, we need to find the ratio between any two consecutive terms of the geometric progression (GP).
Given that the 2nd term is -7 and the 5th term is 56, we can deduce the ratio between consecutive terms as follows:
Ratio = (Value of the 5th term) / (Value of the 2nd term)
= 56 / (-7)
= -8
Therefore, the common ratio of the geometric progression is -8.
To find the first term of the geometric progression, we can use the formula:
First term = (Value of the 2nd term) / (Common ratio)
= -7 / (-8)
= 7/8 or 0.875
Hence, the first term of the geometric progression is 7/8 or 0.875.
To find the sum of the first five terms of the geometric progression, we can use the formula for the sum of a geometric series:
Sum of n terms = (First term * (1 - (Common ratio)^n)) / (1 - Common ratio)
In this case, we want to find the sum of the first five terms, so n = 5.
Sum of first five terms = (0.875 * (1 - (-8)^5)) / (1 - (-8))
= (0.875 * (1 - 32768)) / (1 + 8)
= (0.875 * (-32767)) / 9
= -28671.125 / 9
= -3185.68 (rounded to two decimal places)
Therefore, the sum of the first five terms of the geometric progression is approximately -3185.68.