Advertisements
Advertisements
Question
Find the sum of all natural numbers between 1 and 100, which are divisible by 3.
Advertisements
Solution
In this problem, we need to find the sum of all the multiples of 3 lying between 1 and 100.
So, we know that the first multiple of 3 after 1 is 3 and the last multiple of 3 before 100 is 99.
Also, all these terms will form an A.P. with the common difference of 3.
So here,
First term (a) = 3
Last term (l) = 99
Common difference (d) = 3
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,
99 = 3 + (n - 1)3
99 = 3 + 3n - 3
99 = 3n
Further simplifying
`n = 99/3`
n = 33
Now, using the formula for the sum of n terms,
`S_n = 33/2 [2(3) + (33 - 1)3]`
`= 33/2 [6 + (32)3]`
`= 33/2 (6 + 96)`
`= (33(102))/2`
On further simplification, we get,
`S_n = 33(51)`
= 1683
Therefore, the sum of all the multiples of 3 lying between 1 and 100 is `S_n = 1683`
APPEARS IN
RELATED QUESTIONS
If the sum of the first n terms of an A.P. is `1/2`(3n2 +7n), then find its nth term. Hence write its 20th term.
Determine the A.P. whose 3rd term is 16 and the 7th term exceeds the 5th term by 12.
A contract on a construction job specifies a penalty for delay of completion beyond a certain date as follows: Rs. 200 for the first day, Rs. 250 for the second day, Rs. 300 for the third day, etc., the penalty for each succeeding day being Rs. 50 more than for the preceding day. How much money does the contractor have to pay as a penalty if he has delayed the work by 30 days.
Find the sum of the following arithmetic progressions:
`(x - y)/(x + y),(3x - 2y)/(x + y), (5x - 3y)/(x + y)`, .....to n terms
Find the sum of all odd numbers between 100 and 200.
Find the sum of all integers between 84 and 719, which are multiples of 5.
Find the sum of all even integers between 101 and 999.
The first term of an A.P. is 5, the last term is 45 and the sum of its terms is 1000. Find the number of terms and the common difference of the A.P.
Find the common difference of an AP whose first term is 5 and the sum of its first four terms is half the sum of the next four terms.
How many terms of the AP 63, 60, 57, 54, ….. must be taken so that their sum is 693? Explain the double answer.
The first and the last terms of an A.P. are 8 and 350 respectively. If its common difference is 9, how many terms are there and what is their sum?
Let there be an A.P. with first term 'a', common difference 'd'. If an denotes in nth term and Sn the sum of first n terms, find.
The common difference of the A.P.
If the third term of an A.P. is 1 and 6th term is – 11, find the sum of its first 32 terms.
Find second and third terms of an A.P. whose first term is – 2 and the common difference is – 2.
Find t21, if S41 = 4510 in an A.P.
If the sum of the first m terms of an AP is n and the sum of its n terms is m, then the sum of its (m + n) terms will be ______.
The famous mathematician associated with finding the sum of the first 100 natural numbers is ______.
Find the sum of first 20 terms of an A.P. whose nth term is given as an = 5 – 2n.
Find the sum of first 25 terms of the A.P. whose nth term is given by an = 5 + 6n. Also, find the ratio of 20th term to 45th term.
