If the sum of a geometric series is 781, the number of terms is 5, the common ratio is 5 and the last term is 625, what is the first term a1?
plug in all your info into the formula
( you have extra information)
S(n) = a(r^n -1)/(r-1)
781 = a(5^5 - 1)/(5-1)
781 = a(3124/4)
781 = 781a
a = 1
check: t(5) = 1(5^4) = 625
if the sum of eight terms of a geometric series is
781/640 and the common ratio is 3/4, what is the fourth term of the series?
To find the first term, a₁, of the geometric series, we can use the formula for the sum of a finite geometric series:
Sₙ = a₁(1 - rⁿ) / (1 - r)
Where:
Sₙ is the sum of the first n terms
a₁ is the first term
r is the common ratio
n is the number of terms
Given information:
S₅ = 781
n = 5
r = 5
The last term, a₅, is given as 625.
Now, let's substitute the given values into the formula and solve for a₁:
781 = a₁(1 - 5⁵) / (1 - 5)
Simplify the exponent:
781 = a₁(1 - 3125) / (1 - 5)
Calculate the exponent:
781 = a₁(-3124) / -4
Multiply both sides by -4:
-3124a₁ = 781 * -4
Simplify the equation:
-3124a₁ = -3124
Divide both sides by -3124:
a₁ = -3124 / -3124
a₁ = 1
Therefore, the first term (a₁) of the geometric series is 1.
To find the first term (a1) of a geometric series, we need to use the formula for the sum of a geometric series. The formula is given by:
S = a1 * (1 - r^n) / (1 - r)
Where:
S = Sum of the series
a1 = First term
r = Common ratio
n = Number of terms
In this problem, we are given:
S = 781
n = 5
r = 5
a5 = 625
Using the formula, we can substitute the given values and solve for a1.
781 = a1 * (1 - 5^5) / (1 - 5)
Simplifying the equation,
781 = a1 * (1 - 3125) / -4
781 * -4 = a1 * (-3124)
-3124a1 = -3061844
Now, we can solve for a1 by dividing both sides of the equation by -3124.
a1 = -3061844 / -3124
After dividing, we get:
a1 ≈ 980.919
Therefore, the approximate value of the first term (a1) is 980.919.