Find the sum of the following AP
(a) 2 +4 +6 +....+ 98 +100
(b) (-5) +(-10) +(-15) +...+ (-50)
(a) In this arithmetic progression, the first term is 2, the common difference is 2, and the last term is 100. We can use the formula for the sum of an arithmetic progression:
Sn = (n/2)(a + l)
where Sn is the sum of the arithmetic progression, n is the number of terms, a is the first term, and l is the last term.
In this case, n = (100 - 2)/2 + 1 = 49, a = 2, and l = 100.
Sn = (49/2)(2 + 100)
= (49/2)(102)
= 2499
Therefore, the sum of the arithmetic progression is 2499.
(b) In this arithmetic progression, the first term is -5, the common difference is -5, and the last term is -50. The formula for the sum of an arithmetic progression still applies.
Using the same formula as in part (a):
n = (-50 - (-5))/(-5) + 1 = 10
a = -5
l = -50
Sn = (10/2)(-5 + (-50))
= (10/2)(-55)
= -275
Therefore, the sum of the arithmetic progression is -275.
Let's calculate the sum of each arithmetic progression step by step.
(a) AP: 2, 4, 6, ..., 98, 100
The first term is 2 (a₁ = 2), the common difference is 2 (d = 2), and the last term is 100 (aₙ = 100).
To calculate the sum of an arithmetic progression, we can use the formula:
Sum (S) = (n/2) * (first term + last term)
First, let's find the number of terms (n):
n = (last term - first term) / common difference + 1
= (100 - 2) / 2 + 1
= 98 / 2 + 1
= 49 + 1
= 50
Now, we can substitute the values into the formula:
S = (n/2) * (first term + last term)
= (50/2) * (2 + 100)
= 25 * 102
= 2550
Therefore, the sum of the arithmetic progression 2 + 4 + 6 + ... + 98 + 100 is 2550.
(b) AP: -5, -10, -15, ..., -50
The first term is -5 (a₁ = -5), the common difference is -5 (d = -5), and the last term is -50 (aₙ = -50).
Let's find the number of terms (n):
n = (last term - first term) / common difference + 1
= (-50 - (-5)) / -5 + 1
= (-50 + 5) / -5 + 1
= -45 / -5 + 1
= 9 + 1
= 10
Using the formula, we can find the sum:
S = (n/2) * (first term + last term)
= (10/2) * (-5 + -50)
= 5 * -55
= -275
Therefore, the sum of the arithmetic progression -5 + -10 + -15 + ... + -50 is -275.
To find the sum of an arithmetic progression (AP), you can use the formula:
S = (n/2) * (a + l)
Where:
S is the sum of the AP,
n is the number of terms in the AP,
a is the first term, and
l is the last term.
(a) For the AP 2 + 4 + 6 + .... + 98 + 100:
Here, the first term (a) is 2, the last term (l) is 100, and the common difference is 2. To find the number of terms (n), we can use the formula:
l = a + (n-1)d
Substituting the known values:
100 = 2 + (n-1)2
Simplifying:
98 = 2(n-1)
Dividing both sides by 2:
49 = n - 1
Adding 1 to both sides:
n = 50
Now, we can find the sum (S) using the formula:
S = (n/2) * (a + l)
S = (50/2) * (2 + 100)
S = 25 * 102
S = 2550
Therefore, the sum of the AP 2 + 4 + 6 + .... + 98 + 100 is 2550.
(b) For the AP -5 + (-10) + (-15) + ... + (-50):
Here, the first term (a) is -5, the last term (l) is -50, and the common difference is -5. Again, to find the number of terms (n), we can use the formula:
l = a + (n-1)d
Substituting the known values:
-50 = -5 + (n-1)(-5)
Simplifying:
-50 = -5 -5n + 5
-55 = -5n
Dividing both sides by -5:
11 = n
Now, we can find the sum (S) using the formula:
S = (n/2) * (a + l)
S = (11/2) * (-5 + (-50))
S = (11/2) * (-55)
S = -605
Therefore, the sum of the AP -5 + (-10) + (-15) + ... + (-50) is -605.