Find the first six partial sums S1, S2, S3, S4, S5, S6 of the sequence whose nth term is given.
1
2
,
1
22
,
1
23
,
1
24
, . . .
S1
=
S2
=
S3
=
S4
=
S5
=
S6
=
S1 = 1
S2 = 1 + 2 = 3
S3 = 1 + 2 + 1/22 = 3 + 1/22 = 3.0454 (rounded to four decimal places)
S4 = 1 + 2 + 1/22 + 1/23 = 3 + 1/22 + 1/23 = 3.0904 (rounded to four decimal places)
S5 = 1 + 2 + 1/22 + 1/23 + 1/24 = 3 + 1/22 + 1/23 + 1/24 = 3.1333 (rounded to four decimal places)
S6 = 1 + 2 + 1/22 + 1/23 + 1/24 + 1/25 = 3 + 1/22 + 1/23 + 1/24 + 1/25 = 3.1730 (rounded to four decimal places)
To find the first six partial sums S1, S2, S3, S4, S5, S6 of the sequence with the nth term given as 1/n^2, we can use the formula for the sum of the first n terms of a sequence:
Sn = 1/1^2 + 1/2^2 + 1/3^2 + ... + 1/n^2
Let's calculate the values step-by-step:
S1 = 1/1^2 = 1
S2 = 1/1^2 + 1/2^2 = 1 + 1/4 = 5/4 = 1.25
S3 = 1/1^2 + 1/2^2 + 1/3^2 = 1 + 1/4 + 1/9 = 49/36 ≈ 1.3611
S4 = 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 = 1 + 1/4 + 1/9 + 1/16 = 25/16 ≈ 1.5625
S5 = 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 = 1 + 1/4 + 1/9 + 1/16 + 1/25 = 255/144 ≈ 1.7708
S6 = 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + 1/6^2 = 1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 = 91/36 ≈ 2.5278
Therefore, the first six partial sums S1, S2, S3, S4, S5, S6 of the sequence are:
S1 = 1
S2 = 1.25
S3 ≈ 1.3611
S4 ≈ 1.5625
S5 ≈ 1.7708
S6 ≈ 2.5278