The sum Sn of the first n terms of a sequence is given by Sn=n(n+2).
Find:
(¡) the sum of the twentieth,twenty-first and twenty-second terms of the sequence;
(¡¡) the first three terms of the sequence.
(¡) To find the sum of the twentieth, twenty-first, and twenty-second terms of the sequence, we substitute n with 20, 21, and 22 in the formula Sn=n(n+2).
For the twentieth term (n=20):
S20 = 20(20+2)
S20 = 20(22)
S20 = 440
For the twenty-first term (n=21):
S21 = 21(21+2)
S21 = 21(23)
S21 = 483
For the twenty-second term (n=22):
S22 = 22(22+2)
S22 = 22(24)
S22 = 528
To find the sum of these three terms, we add them up:
440 + 483 + 528 = 1451
So, the sum of the twentieth, twenty-first, and twenty-second terms of the sequence is 1451.
(¡¡) To find the first three terms of the sequence, we substitute n with 1, 2, and 3 in the formula Sn=n(n+2).
For the first term (n=1):
S1 = 1(1+2)
S1 = 1(3)
S1 = 3
For the second term (n=2):
S2 = 2(2+2)
S2 = 2(4)
S2 = 8
For the third term (n=3):
S3 = 3(3+2)
S3 = 3(5)
S3 = 15
So, the first three terms of the sequence are 3, 8, and 15.
To find the sum of the twentieth, twenty-first, and twenty-second terms of the sequence, we need to substitute the values into the given formula Sn = n(n + 2).
(i) Sum of the twentieth, twenty-first, and twenty-second terms:
When n = 20:
S20 = 20(20 + 2) = 20(22) = 440
When n = 21:
S21 = 21(21 + 2) = 21(23) = 483
When n = 22:
S22 = 22(22 + 2) = 22(24) = 528
Therefore, the sum of the twentieth, twenty-first, and twenty-second terms of the sequence is 440 + 483 + 528 = 1451.
(ii) First three terms of the sequence:
To find the first three terms, substitute n = 1, 2, and 3 into the formula.
When n = 1:
S1 = 1(1 + 2) = 1(3) = 3
When n = 2:
S2 = 2(2 + 2) = 2(4) = 8
When n = 3:
S3 = 3(3 + 2) = 3(5) = 15
Therefore, the first three terms of the sequence are 3, 8, and 15.
To find the sum of the twentieth, twenty-first, and twenty-second terms of the sequence, we need to substitute the given values into the formula and simplify the expression.
(¡) Let's calculate the sum of the twentieth, twenty-first, and twenty-second terms of the sequence.
The sum of the first n terms, Sn, is given by Sn = n(n + 2).
Substituting n = 20, we have S20 = 20(20 + 2).
Simplifying further, S20 = 20(22) = 440.
So, the sum of the twentieth, twenty-first, and twenty-second terms of the sequence is 440.
(¡¡) To find the first three terms of the sequence, we can use the formula Sn = n(n + 2) again.
The sum of the first n terms, Sn, is given by Sn = n(n + 2).
Substituting n = 1, we have S1 = 1(1 + 2).
Simplifying further, S1 = 1(3) = 3.
So, the first term of the sequence is 3.
To find the second term, we need to subtract the sum of the first (n - 1) terms from the sum of the first n terms. In this case, n = 2.
S2 = S2 - S1
= 2(2 + 2) - 3
= 8 - 3
= 5.
So, the second term of the sequence is 5.
Similarly, to find the third term, we subtract the sum of the first (n - 1) terms from the sum of the first n terms. In this case, n = 3.
S3 = S3 - S2
= 3(3 + 2) - 8
= 15 - 8
= 7.
Therefore, the first three terms of the sequence are 3, 5, and 7.
(i) S22 - S19 = 22*23 - 19*22 = 4*22 = 88
(ii) S1 = 1*3 = 3 = 1st term
S2 = 2*4 = 8 = 1st + 2nd = 3 + 2nd so the 2nd term is 5
S3 = 3*5 = 15 so the 3rd term is 7