Find the first six partial sums S1, S2, S3, S4, S5, S6 of the sequence.
1/2, 1/2^2, 1/2^3, 1/2^4'...
an = ( 1 / 2 ) ^ n
a1 = 1 / 2
a2 = 1 / 2 ^ 2 = 1 / 4
a3 = 1 / 2 ^ 3 = 1 / 8
a4 = 1 / 2 ^ 4 = 1 / 16
a5 = 1 / 2 ^ 5 = 1 / 32
a6 = 1 / 2 ^ 6 = 1 / 64
1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + 1 / 32 + 1 / 64 = 63 / 64
To find the first six partial sums of the sequence 1/2, 1/2^2, 1/2^3, 1/2^4, and so on, we need to add up each term of the sequence in order. Here are the calculations:
S1 = 1/2
S2 = (1/2) + (1/2^2) = 1/2 + 1/4 = 3/4
S3 = (1/2) + (1/2^2) + (1/2^3) = 1/2 + 1/4 + 1/8 = 7/8
S4 = (1/2) + (1/2^2) + (1/2^3) + (1/2^4) = 1/2 + 1/4 + 1/8 + 1/16 = 15/16
S5 = (1/2) + (1/2^2) + (1/2^3) + (1/2^4) + (1/2^5) = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 = 31/32
S6 = (1/2) + (1/2^2) + (1/2^3) + (1/2^4) + (1/2^5) + (1/2^6) = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 = 63/64
Therefore, the first six partial sums of the sequence are S1 = 1/2, S2 = 3/4, S3 = 7/8, S4 = 15/16, S5 = 31/32, and S6 = 63/64.
To find the partial sums of the sequence, we need to add up the terms of the sequence in a cumulative manner.
Let's start by writing out the terms of the sequence:
1/2, 1/2^2, 1/2^3, 1/2^4, ...
To find S1, the first partial sum, we add up the first term of the sequence:
S1 = 1/2
To find S2, the second partial sum, we add up the first two terms of the sequence:
S2 = 1/2 + 1/2^2 = 1/2 + 1/4 = 3/4
To find S3, the third partial sum, we add up the first three terms of the sequence:
S3 = 1/2 + 1/2^2 + 1/2^3 = 1/2 + 1/4 + 1/8 = 7/8
To find S4, the fourth partial sum, we add up the first four terms of the sequence:
S4 = 1/2 + 1/2^2 + 1/2^3 + 1/2^4 = 1/2 + 1/4 + 1/8 + 1/16 = 15/16
To find S5, the fifth partial sum, we add up the first five terms of the sequence:
S5 = 1/2 + 1/2^2 + 1/2^3 + 1/2^4 + 1/2^5 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 = 31/32
To find S6, the sixth partial sum, we add up the first six terms of the sequence:
S6 = 1/2 + 1/2^2 + 1/2^3 + 1/2^4 + 1/2^5 + 1/2^6 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 = 63/64
So, the first six partial sums of the sequence are:
S1 = 1/2
S2 = 3/4
S3 = 7/8
S4 = 15/16
S5 = 31/32
S6 = 63/64