Advertisements
Advertisements
Question
Find the sum 2 + 4 + 6 ... + 200
Advertisements
Solution
In the given problem, we need to find the sum of terms for different arithmetic progressions. So, here we use the following formula for the sum of n terms of an A.P.,
`S_n = n/2[2a + (n -1)d]`
Where; a = first term for the given A.P.
d = common difference of the given A.P.
n = number of terms
2 + 4 + 6 ... + 200
Common difference of the A.P. (d) = `a_2 - a_1`
= 6 - 4
= 2
So here,
First term (a) = 2
Last term (l) = 200
Common difference (d) = 2
So, here the first step is to find the total number of terms. Let us take the number of terms as n.
Now, as we know,
`a_n = a + (n -1)d`
So, for the last term,
200 = 2 +(n - 1)2
200 = 2 + 2n - 2
200 = 2n
Further simplifying,
`n = 200/2`
n = 100
Now, using the formula for the sum of n terms, we get
`S_n = 100/2 [2(2) + (100 - 1)2]`
= 50 [4 + (99)2]
= 50(4 + 198)
On further simplification we get
`S_n = 50(202)`
= 10100
Therefore, the sum of the A.P is `S_n = 10100`
APPEARS IN
RELATED QUESTIONS
The ratio of the sums of m and n terms of an A.P. is m2 : n2. Show that the ratio of the mth and nth terms is (2m – 1) : (2n – 1)
Find the sum of the following arithmetic progressions: 50, 46, 42, ... to 10 terms
Find the sum of the following arithmetic progressions:
a + b, a − b, a − 3b, ... to 22 terms
Find the sum of all integers between 100 and 550, which are divisible by 9.
If numbers n – 2, 4n – 1 and 5n + 2 are in A.P., find the value of n and its next two terms.
How many three-digit natural numbers are divisible by 9?
Write the next term for the AP` sqrt( 8), sqrt(18), sqrt(32),.........`
Find the sum of the first n natural numbers.
How many terms of the AP `20, 19 1/3 , 18 2/3, ...` must be taken so that their sum is 300? Explain the double answer.
In an A.P. 19th term is 52 and 38th term is 128, find sum of first 56 terms.
If first term of an A.P. is a, second term is b and last term is c, then show that sum of all terms is \[\frac{\left( a + c \right) \left( b + c - 2a \right)}{2\left( b - a \right)}\].
For what value of n, the nth terms of the arithmetic progressions 63, 65, 67, ... and 3, 10, 17, ... equal?
The sum of first seven terms of an A.P. is 182. If its 4th and the 17th terms are in the ratio 1 : 5, find the A.P.
Write the expression of the common difference of an A.P. whose first term is a and nth term is b.
Find second and third terms of an A.P. whose first term is – 2 and the common difference is – 2.
Write the formula of the sum of first n terms for an A.P.
In an AP if a = 1, an = 20 and Sn = 399, then n is ______.
Rohan repays his total loan of ₹ 1,18,000 by paying every month starting with the first installment of ₹ 1,000. If he increases the installment by ₹ 100 every month, what amount will be paid by him in the 30th installment? What amount of loan has he paid after 30th installment?
The nth term of an A.P. is 6n + 4. The sum of its first 2 terms is ______.
