Advertisements
Advertisements
प्रश्न
Find the sum of all 3-digit natural numbers, which are multiples of 11.
Advertisements
उत्तर
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
संबंधित प्रश्न
The houses in a row numbered consecutively from 1 to 49. Show that there exists a value of x such that sum of numbers of houses preceding the house numbered x is equal to sum of the numbers of houses following x.
If the term of m terms of an A.P. is the same as the sum of its n terms, show that the sum of its (m + n) terms is zero
An A.P. consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term of the A.P.
Find the sum of first 15 multiples of 8.
A sum of Rs 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs 20 less than its preceding prize, find the value of each of the prizes.
Find the sum of n terms of an A.P. whose nth terms is given by an = 5 − 6n.
Find the middle term of the AP 6, 13, 20, …., 216.
Find the sum of first n terms of an AP whose nth term is (5 - 6n). Hence, find the sum of its first 20 terms.
If the common differences of an A.P. is 3, then a20 − a15 is
There are 25 rows of seats in an auditorium. The first row is of 20 seats, the second of 22 seats, the third of 24 seats, and so on. How many chairs are there in the 21st row ?
Fill up the boxes and find out the number of terms in the A.P.
1,3,5,....,149 .
Here a = 1 , d =b`[ ], t_n = 149`
tn = a + (n-1) d
∴ 149 =`[ ] ∴149 = 2n - [ ]`
∴ n =`[ ]`
Q.16
Show that a1, a2, a3, … form an A.P. where an is defined as an = 3 + 4n. Also find the sum of first 15 terms.
Find the common difference of an A.P. whose first term is 5 and the sum of first four terms is half the sum of next four terms.
Find the sum of first 1000 positive integers.
Activity :- Let 1 + 2 + 3 + ........ + 1000
Using formula for the sum of first n terms of an A.P.,
Sn = `square`
S1000 = `square/2 (1 + 1000)`
= 500 × 1001
= `square`
Therefore, Sum of the first 1000 positive integer is `square`
Find the sum of three-digit natural numbers, which are divisible by 4.
First four terms of the sequence an = 2n + 3 are ______.
The sum of first n terms of the series a, 3a, 5a, …….. is ______.
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.
