Find the sum of the first 12 terms of this sequence:
9,4.5,2.25,...
I believe that the pattern is dividing the term by 2,and I keep getting 7.9956 for the sum, but that is incorrect.
Anyone understand this?
I might be wrong, but i got 17.99121094.
I just calculated again and got that. Thank you!
Geometric Sequences :
xn = a * r ^ ( n - 1 )
In this case :
a = 9 , r = 1 / 2
xn = a * ( 1 / 2 ) ^ ( n - 1 )
Sum:
S = a ( 1 - r ^ n ) / ( 1 - r )
a = 9 , r = 1 /8 2 , n = 12 so :
S = 9 * [ 1 - ( 1 / 2 ) ^ 12 ] / ( 1 - 1 / 2 ) =
9 * ( 1 - 1 / 4096 ) / ( 1 / 2 ) =
2 * 9 * ( 4096 / 4096 - 1 / 4096 ) =
18 * 4095 / 4096 =
73710 / 4096 =
17.99560546875
a = 9 , r = 1 / 2 , n = 12
To find the sum of the first 12 terms of the given sequence, we can use the formula for the sum of a geometric series.
The formula for the sum of a geometric series is:
S = a * (r^n - 1) / (r - 1)
Where:
S is the sum of the series
a is the first term of the series
r is the common ratio
n is the number of terms
In this case, the first term (a) is 9, the common ratio (r) is 1/2, and the number of terms (n) is 12.
Substituting these values into the formula, we get:
S = 9 * (1/2^12 - 1) / (1/2 - 1)
Simplifying this equation further:
S = 9 * (1/2^12 - 1) / (-1/2)
S = 9 * (-1/4096 - 1) / (-1/2)
S = 9 * (-1/4096 - 4096/4096) / (-1/2)
S = 9 * (-4097/4096) / (-1/2)
S = 9 * (-4097/4096) * (-2/1)
S = 9 * 8194 / 4096
S = 18
Therefore, the sum of the first 12 terms of the given sequence is 18.
It looks like your calculation of 7.9956 was close, but you may have made a rounding error along the way.