determine the number of terms n in each geometric series
21) a_1 = -2, r = 5, S_n = -62
23) a_1 = -3, r = 4, S_n = -409
thank u! :-)
I'll do the first, you do the 2nd in the same way
a = -2
r = 5
Sum(n) = a(r^n - 1)/(r-1)
-62 = -2(5^n - 1)/4
124 = 5^n - 1
5^n = 125
by observation, n = 3
long way if it were not so obvious,
log 5^n = log 125
n log 5 = log 125
n = log 125/log5 = 3
take log of both sides
Sure! Let's find the value of n for each geometric series.
21) For the first series with a_1 = -2, r = 5, and S_n = -62.
The formula for the sum of the first n terms of a geometric series is given by:
S_n = a_1 * (1 - r^n) / (1 - r)
Substituting the given values, we have:
-62 = -2 * (1 - 5^n) / (1 - 5)
Simplifying further:
-62 = (-2 * (1 - 5^n)) / -4
Multiply both sides by -4:
248 = 2 * (1 - 5^n)
Divide both sides by 2:
124 = 1 - 5^n
Rearranging the equation:
5^n = 1 - 124
5^n = -123
Since the right side is negative, there are no values of n that would satisfy this equation. Therefore, there are no terms in this geometric series.
23) For the second series with a_1 = -3, r = 4, and S_n = -409.
Using the same formula,
S_n = a_1 * (1 - r^n) / (1 - r)
Substituting the given values,
-409 = -3 * (1 - 4^n) / (1 - 4)
Simplifying,
-409 = (-3 * (1 - 4^n)) / -3
Multiply both sides by -3,
1227 = 3 * (1 - 4^n)
Divide both sides by 3,
409 = 1 - 4^n
Rearranging the equation,
4^n = 1 - 409
4^n = -408
Since the right side is negative, there are no values of n that would satisfy this equation. Therefore, there are no terms in this geometric series either.
In both cases, there are no terms in the given geometric series.
To determine the number of terms (n) in each geometric series, we can use the formula for the sum of an infinite geometric series:
S_n = a_1 / (1 - r)
1) For the first geometric series:
a_1 = -2
r = 5
S_n = -62
Plugging these values into the formula:
-62 = -2 / (1 - 5)
Simplifying the equation gives:
-62 = -2 / (-4)
To solve for n, we need to determine the value of (-2 / (-4)). Dividing -2 by -4 gives 0.5.
Therefore, the equation becomes:
-62 = 0.5
This equation is not true, which means there is no solution. There are no terms in this geometric series that satisfy these conditions.
2) For the second geometric series:
a_1 = -3
r = 4
S_n = -409
Plugging these values into the formula:
-409 = -3 / (1 - 4)
Simplifying the equation gives:
-409 = -3 / (-3)
To solve for n, we need to determine the value of (-3 / (-3)). Dividing -3 by -3 gives 1.
Therefore, the equation becomes:
-409 = 1
This equation is also not true, which means there is no solution. There are no terms in this geometric series that satisfy these conditions.
In both cases, there are no terms in the given geometric series that produce the given sum.