Find the sum given for each of the following series:(a) Sn=1+5+9+....(n=15)
(b) Sn=(-8)+(-3)+2+... (n=20)
you have
a = 1
d = 4
So, S15 = 15/2 (2*1 + 14*4)
For (b) , you have
a = -8
d = 5
And as always,
Sn = n/2 (2a + (n-1)d)
Now plug in your numbers
(a) the common difference is 4 ... the 15th term is ... 1 + [4 * (15 - 1)] = 57
... S15 = (1 + 57) * (15 / 2)
(b) the common difference is 5 ... the 20th term is ... -8 + [5 * (20 - 1)] = 87
... S20 = (-8 + 87) * (20 / 2)
(a) To find the sum of the series Sn = 1 + 5 + 9 + ... (n=15), we need to identify the pattern of the series.
In this case, we can observe that the series is an arithmetic series with a common difference of 4.
To find the sum of an arithmetic series, we can use the formula:
Sn = (n/2) * (first term + last term)
For this series, the first term (a) is 1, and the last term (l) can be found using the formula for the nth term of an arithmetic series:
l = a + (n-1)d
where d is the common difference. In this case, d = 4.
Calculating the last term:
l = 1 + (15-1) * 4
= 1 + 14 * 4
= 1 + 56
= 57
Plugging the values into the sum formula:
Sn = (15/2) * (1 + 57)
= 7.5 * 58
= 435
Therefore, the sum of the series given is 435.
(b) To find the sum of the series Sn = -8 + (-3) + 2 + ... (n=20), we can again identify that the series is arithmetic with a common difference of 5.
Using the same formulas as before, we can find the first term (a), the last term (l), and the sum (Sn) of the series.
Calculating the last term:
l = -8 + (20-1) * 5
= -8 + 19 * 5
= -8 + 95
= 87
Plugging the values into the sum formula:
Sn = (20/2) * (-8 + 87)
= 10 * 79
= 790
Therefore, the sum of the series given is 790.
To find the sum of each series, we can use the concept of arithmetic progression.
(a) For the series Sn = 1 + 5 + 9 + ... (n = 15):
The terms in this series have a common difference of 4, i.e., each term is obtained by adding 4 to the previous term.
Step 1: Find the nth term of the series.
The nth term of an arithmetic progression is given by the formula:
an = a1 + (n - 1)d
where a1 is the first term, d is the common difference, and n is the position of the term.
In this case, a1 = 1, d = 4, and n = 15.
an = 1 + (15 - 1) * 4
an = 1 + 14 * 4
an = 1 + 56
an = 57
Step 2: Find the sum of the series using the formula for the sum of an arithmetic series.
The sum of an arithmetic progression is given by the formula:
Sn = (n/2)(a1 + an)
In this case, n = 15, a1 = 1, and an = 57.
Sn = (15/2)(1 + 57)
Sn = (15/2)(58)
Sn = 435
Therefore, the sum of the series is 435.
(b) For the series Sn = (-8) + (-3) + 2 + ... (n = 20):
The terms in this series have a common difference of 5, i.e., each term is obtained by adding 5 to the previous term.
Step 1: Find the nth term of the series.
Using the same formula as in part (a), we can find the nth term:
an = a1 + (n - 1)d
In this case, a1 = -8, d = 5, and n = 20.
an = -8 + (20 - 1) * 5
an = -8 + 19 * 5
an = -8 + 95
an = 87
Step 2: Find the sum of the series using the formula for the sum of an arithmetic series.
Again, we use the formula:
Sn = (n/2)(a1 + an)
In this case, n = 20, a1 = -8, and an = 87.
Sn = (20/2)(-8 + 87)
Sn = (10)(79)
Sn = 790
Therefore, the sum of the series is 790.