In a geometric series S4 = 64 and r = -2/3. The first term in the sequence is?
Thanks for your help!
To find the first term in the geometric series, we can use the formula for the sum of a geometric series:
Sn = a * (1 - r^n) / (1 - r)
Where:
Sn is the sum of n terms
a is the first term
r is the common ratio
In this case, we are given that:
S4 = 64, which is the sum of 4 terms
r = -2/3
Substituting these values into the formula, we have:
64 = a * (1 - (-2/3)^4) / (1 - (-2/3))
To simplify this expression, we need to calculate (-2/3)^4:
(-2/3)^4 = (-2/3) * (-2/3) * (-2/3) * (-2/3) = 16/81
Substituting this value back into the equation, we have:
64 = a * (1 - 16/81) / (1 + 2/3)
To simplify further, let's find the common denominator of 81 and 3:
(1 - 16/81) / (1 + 2/3) = (81/81 - 16/81) / (3/3 + 2/3)
= (65/81) / (5/3)
= (65/81) * (3/5)
= 195/405
Now we can rewrite the equation as:
64 = a * (195/405)
To solve for a, we can multiply both sides by (405/195):
64 * (405/195) = a
Simplifying:
a ≈ 132.308
Therefore, the first term in the geometric series is approximately 132.308.
Is Sn the nth sum or the nth term?
If S4 is the 4th TERM, then
S3 = (-3/2)S4 = (-3/2)*64 = -96
S2 = (-3/2)*S3 = 144
S1 = -216
If S4 is the sum of the sequence of 4 terms, then you have to solve a more complicated equation to get S1.