the sum of all even numbers between 100 and 200 is 7550
Step-by-step explanation:
To find the sum of all even numbers between 100 and 200 using an arithmetic progression, we can set up the series with the first term (a) as 102 (the first even number after 100), the common difference (d) as 2 (since we are adding even numbers), and the last term (l) as 200.
The nth term of an arithmetic progression can be calculated using the formula: an = a + (n - 1) * d.
To find the number of terms (n), we can use the formula: n = (l - a) / d + 1.
Now let's plug in the values and calculate the sum.
a = 102
d = 2
l = 200
n = (l - a) / d + 1
= (200 - 102) / 2 + 1
= 98 / 2 + 1
= 49 + 1
= 50
So, there are 50 terms in the series.
The sum of an arithmetic series can be calculated using the formula: Sn = (n/2) * (2a + (n - 1) * d).
Substituting the values into the formula, we have:
Sn = (n/2) * (2a + (n - 1) * d)
= (50/2) * (2 * 102 + (50 - 1) * 2)
= 25 * (204 + 49 * 2)
= 25 * (204 + 98)
= 25 * 302
= 7550
Therefore, the sum of all even numbers between 100 and 200 is 7550.
Answers & Comments
Answer-
Given-
a (1st term of an AP) = 100
[tex]\sf \: a_{n}[/tex](last term of an AP = 200
d (Common difference) = 2
To find-
Solution-
To finding the sum of an AP , firstly found the n ( number of terms ) in an AP
Using formula,
[tex]\boxed{ \sf \leadsto \: a_{n} = a + (n - 1)d} \\[/tex]
[tex]\implies[/tex] 200 = 100 + (n-1)2
[tex]\implies[/tex] 200 - 100 = (n-1)2
[tex]\implies[/tex] 100 ÷ 2 = (n-1)
[tex]\implies[/tex] 50 + 1 = n
[tex]\boxed{ \sf \leadsto n \: = 51} \\[/tex]
Number of terms in an AP is 51
For finding sum of an AP
Using formula,
[tex]\boxed{ \sf \leadsto S_{n} = \frac{n}{2} (2a + (n - 1)d)} \\[/tex]
[tex] \sf \implies \: S_{n} \: = \frac{51}{2} (200 + (51 - 1)2) \\ \\[/tex]
[tex]\sf \implies \: S_{n} \: = \frac{51}{2} (200 + 100) \\ \\[/tex]
[tex]\sf \implies \: S_{n} \: = \frac{51}{2} \times 300 \\ \\[/tex]
[tex]\sf \implies \: S_{n} \: = \: 51 \: \times 150 \\ \\[/tex]
[tex]\boxed {\bf \leadsto \: S_{n} \: = 7650}[/tex]
Hence , the sum of all even numbers between 100 and 200 is 7650
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Answer:
the sum of all even numbers between 100 and 200 is 7550
Step-by-step explanation:
To find the sum of all even numbers between 100 and 200 using an arithmetic progression, we can set up the series with the first term (a) as 102 (the first even number after 100), the common difference (d) as 2 (since we are adding even numbers), and the last term (l) as 200.
The nth term of an arithmetic progression can be calculated using the formula: an = a + (n - 1) * d.
To find the number of terms (n), we can use the formula: n = (l - a) / d + 1.
Now let's plug in the values and calculate the sum.
a = 102
d = 2
l = 200
n = (l - a) / d + 1
= (200 - 102) / 2 + 1
= 98 / 2 + 1
= 49 + 1
= 50
So, there are 50 terms in the series.
The sum of an arithmetic series can be calculated using the formula: Sn = (n/2) * (2a + (n - 1) * d).
Substituting the values into the formula, we have:
Sn = (n/2) * (2a + (n - 1) * d)
= (50/2) * (2 * 102 + (50 - 1) * 2)
= 25 * (204 + 49 * 2)
= 25 * (204 + 98)
= 25 * 302
= 7550
Therefore, the sum of all even numbers between 100 and 200 is 7550.