Find the first six partial sums S1, S2, S3, S4, S5, S6 of the sequence.
2, 5, 8, 11, . . .
The sum of the members of a arithmetic progression :
Sn = ( n / 2 ) * [ 2 a1 + ( n - 1 ) * d ]
d = common difference of successive members of arithmetic progression
In this case :
S1 = ( 1 / 2 ) * [ 2 * 2 + ( 1 - 1 ) * 3 ]
S1 = ( 1 / 2 ) * (4 + 0 * 3 )
S1 = ( 1 / 2 ) * 4 = 2
S2 = ( 2 / 2 ) * [ 2 * 2 + ( 2 - 1 ) * 3 ]
S2 = 1 * ( 4 + 1 * 3 )
S2 = 4 + 3 = 7
S3 = ( 3 / 2 ) * [ 2 * 2 + ( 3 - 1 ) * 3 ]
S3 = ( 3 / 2 ) * ( 4 + 2 * 3 )
S3 = ( 3 / 2 ) * ( 4 + 6 )
S3 = ( 3 / 2 ) * 10
S3 = 30 / 2 = 15
S4 = ( 4 / 2 ) * [ 2 * 2 + ( 4 - 1 ) * 3 ]
S4 = 2 * ( 4 + 3 * 3 )
S4 = 2 * ( 4 + 9 )
S4 = 2 * 13
S4 = 26
S5 = ( 5 / 2 ) * [ 2 * 2 + ( 5 - 1 ) * 3 ]
S5 = ( 5 / 2 ) * ( 4 + 4 * 3 )
S5 = ( 5 / 2 ) * ( 4 + 12 )
S5 = ( 5 / 2 ) * 16
S5 = 80 / 2 = 40
S6 = ( 6 / 2 ) * [ 2 * 2 + ( 6 - 1 ) * 3 ]
S6 = 3 * ( 4 + 5 * 3 )
S6 = 3 * ( 4 + 15 )
S6 = 3 * 19
S6 = 57
To find the first six partial sums of the sequence 2, 5, 8, 11, ...
We have the terms of the sequence:
a1 = 2
a2 = 5
a3 = 8
a4 = 11
...
The partial sums S1, S2, S3, S4, S5, S6 can be calculated as follows:
S1 = a1 = 2
S2 = a1 + a2 = 2 + 5 = 7
S3 = a1 + a2 + a3 = 2 + 5 + 8 = 15
S4 = a1 + a2 + a3 + a4 = 2 + 5 + 8 + 11 = 26
S5 = a1 + a2 + a3 + a4 + a5 = 2 + 5 + 8 + 11 + ... = 2 + 5 + 8 + 11 + 14 = 40
S6 = a1 + a2 + a3 + a4 + a5 + a6 = 2 + 5 + 8 + 11 + 14 + ... = 2 + 5 + 8 + 11 + 14 + 17 = 57
Therefore, the first six partial sums of the sequence 2, 5, 8, 11, ... are:
S1 = 2
S2 = 7
S3 = 15
S4 = 26
S5 = 40
S6 = 57
To find the partial sums of a sequence, you add up the terms of the sequence up to a certain position. In this case, we have the sequence:
2, 5, 8, 11, . . .
To find the first partial sum (S1), we add up the first term of the sequence:
S1 = 2
To find the second partial sum (S2), we add up the first two terms of the sequence:
S2 = 2 + 5 = 7
To find the third partial sum (S3), we add up the first three terms of the sequence:
S3 = 2 + 5 + 8 = 15
To find the fourth partial sum (S4), we add up the first four terms of the sequence:
S4 = 2 + 5 + 8 + 11 = 26
To find the fifth partial sum (S5), we add up the first five terms of the sequence:
S5 = 2 + 5 + 8 + 11 + 14 = 40
To find the sixth partial sum (S6), we add up the first six terms of the sequence:
S6 = 2 + 5 + 8 + 11 + 14 + 17 = 59
So, the first six partial sums of the sequence are:
S1 = 2
S2 = 7
S3 = 15
S4 = 26
S5 = 40
S6 = 59