For the geometric series given by 2+4+8+... , which of the following statements is FALSE?
Select one:
a. S200>S199
b. S200>a200
c. S1=a1
d. None of the other 3 statements here are False.
In a geometric sequence:
an = a1 ∙ rⁿ⁻¹
Sum of the first n terms of a geometric sequence:
Sn = a1 ∙ ( 1 − rⁿ ) / ( 1− r )
where
a1 = first term
r is the common ratio
In tis case
a1 = 2 , r = 2
so
Sn = a1 ∙ ( 1 − rⁿ ) / ( 1− r )
S199 = 2 ∙ ( 1 − 2¹⁹⁹ ) / ( 1− 2 )
S199 = 2 ∙ ( 1 − 2¹⁹⁹ ) / ( - 1 )
S199 = - 2 ∙ ( 1 − 2¹⁹⁹ ) = 1.6069380442589902755419620923412 ∙ 10⁶⁰
S200 = 2 ∙ ( 1 − 2²⁰⁰ ) / ( 1− 2 )
S200 = 2 ∙ ( 1 − 2²⁰⁰ ) / ( - 1 )
S200 = - 2 ∙ ( 1 − 2²⁰⁰ ) = 3.2138760885179805510839241846823 ∙ 10⁶⁰
S200 > S199
3.2138760885179805510839241846823 ∙ 10⁶⁰ > 1.6069380442589902755419620923412 ∙ 10⁶⁰
True
an = a ∙ rⁿ⁻¹
a200 = 2 ∙ 2¹⁹⁹ = 2²⁰⁰ = 1.6069380442589902755419620923412 ∙ 10⁶⁰
S200 > a200
3.2138760885179805510839241846823 ∙ 10⁶⁰ > 1.6069380442589902755419620923412 ∙ 10⁶⁰
True
S1 = a1
2 = 2
True
d. None of the other 3 statements here are False.
Well, let's break it down.
a. S200 > S199: This statement is true. As we add more terms to the series, the sum increases, so S200 will be greater than S199.
b. S200 > a200: This statement is also true. The sum S200 will definitely be greater than the 200th term of the series.
c. S1 = a1: Ah, here we have a false statement! The first term of the series is 2, but the sum of the series S1 will also be 2 because it's only the first term.
So, the false statement is c. S1 = a1.
To determine which statement is false, let's analyze each option:
a. S200 > S199: This statement is true. In a geometric series, each term is obtained by multiplying the previous term by a common ratio. As the terms increase, the sum of the series also increases. Therefore, S200 will indeed be greater than S199.
b. S200 > a200: This statement is also true. In a geometric series, the nth term can be calculated using the formula a * r^(n-1), where a is the first term, r is the common ratio, and n is the term number. Since the sum of a geometric series is calculated using the formula S = a * (1 - r^n) / (1 - r), we can see that S200 will be larger than a200.
c. S1 = a1: This statement is true. In a geometric series, the first term of the series is equal to the sum of the series until that point. Therefore, S1 will equal a1.
Based on the analysis above, all three statements are true, which means the answer is:
d. None of the other 3 statements here are false.
To determine which statement is false among the given options for the geometric series, let's break down each statement and analyze them.
a. S200 > S199:
To find the sum of a geometric series, we apply the formula: Sn = a * (r^n - 1) / (r - 1), where "S" denotes the sum of the series, "n" represents the number of terms, "a" is the first term, and "r" is the common ratio.
In this case, the first term (a) is 2, the common ratio (r) is 2, and we need to compare S200 (the sum of the first 200 terms) with S199 (the sum of the first 199 terms).
Using the formula, we find that S200 = 2 * (2^200 - 1) / (2 - 1) and S199 = 2 * (2^199 - 1) / (2 - 1).
Since both S200 and S199 have the same multiplying factor (2/1) and differ only in the exponent of 2, S200 will always be greater than S199. Therefore, statement a is True, and not false.
b. S200 > a200:
Here, we compare the sum of the first 200 terms (S200) with the 200th term (a200) of the series.
To calculate a specific term of a geometric series, we use the formula: an = a * r^(n-1), where "an" represents the nth term.
In this case, the first term (a) is 2, the common ratio (r) is 2, and we want to know if S200 is greater than a200.
S200 = 2 * (2^200 - 1) / (2 - 1) and a200 = 2 * 2^(200-1).
By simplifying both expressions, we can see that S200 equals 2 * (2^199) and a200 equals 2 * (2^199).
Since they are equal, the statement is false.
c. S1 = a1:
To check if the sum of the first term (S1) is equal to the first term (a1) of the geometric series, we compare them.
S1 = a * (r^1 - 1) / (r - 1) and a1 is simply the first term of the series.
In this case, a1 = 2 and S1 = 2 * (2^1 - 1) / (2 - 1).
After simplification, we find that S1 equals 2, which is equal to a1. Therefore, the statement is true.
d. None of the other 3 statements here are false:
Based on our analysis, we have found that statement b is false (S200 is not greater than a200).
Therefore, statement d is also false because b is among the given statements.
In conclusion, the false statement among the given options is statement b: S200 > a200.