Advertisements
Advertisements
Question
Find the sum of all 3-digit natural numbers, which are multiples of 11.
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
all 3-digit natural numbers, which are multiples of 11.
We know that the first 3 digit number multiple of 11 will be 110.
Last 3 digit number multiple of 11 will be 990.
So here,
First term (a) = 110
Last term (l) = 990
Common difference (d) = 11
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
990 = 110 + (n -1) 11
990 = 110 + 11n - 11
990 = 99 + 11n
891 = 11n
81 = n
Now, using the formula for the sum of n terms, we get
`S_n = 81/2 [2(110) + (81 - 1)11]`
`S_n = 81/2 [220 + 80 xx 11]`
`S_n = 81/2 xx 1100`
`S_n = 81 xx 550`
`S_n = 44550`
Therefore, the sum of all the 3 digit multiples of 11 is 44550.
APPEARS IN
RELATED QUESTIONS
If the sum of the first n terms of an AP is 4n − n2, what is the first term (that is, S1)? What is the sum of the first two terms? What is the second term? Similarly, find the 3rd, the 10th, and the nth terms.
Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.
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 3 - digit natural numbers which are divisible by 13.
Determine the A.P. Whose 3rd term is 16 and the 7th term exceeds the 5th term by 12.
How many terms of the A.P. : 24, 21, 18, ................ must be taken so that their sum is 78?
How many two-digits numbers are divisible by 3?
If the 9th term of an A.P. is zero then show that the 29th term is twice the 19th term?
The sequence −10, −6, −2, 2, ... is ______.
There are 37 terms in an A.P., the sum of three terms placed exactly at the middle is 225 and the sum of last three terms is 429. Write the A.P.
The sum of third and seventh term of an A. P. is 6 and their product is 8. Find the first term and the common difference of the A. P.
Find the sum of all 2 - digit natural numbers divisible by 4.
Let the four terms of the AP be a − 3d, a − d, a + d and a + 3d. find A.P.
Q.10
Q.13
The famous mathematician associated with finding the sum of the first 100 natural numbers is ______.
If the nth term of an AP is (2n +1), then the sum of its first three terms is ______.
In an AP, if Sn = n(4n + 1), find the AP.
Find the sum of first 17 terms of an AP whose 4th and 9th terms are –15 and –30 respectively.
Sum of 1 to n natural number is 45, then find the value of n.
