Advertisements
Advertisements
Question
Find the sum of the integers between 100 and 200 that are
- divisible by 9
- not divisible by 9
[Hint (ii) : These numbers will be : Total numbers – Total numbers divisible by 9]
Advertisements
Solution
i. The numbers (integers) between 100 and 200 which is divisible by 9 are 108, 117, 126,..., 198.
Let n be the number of terms between 100 and 200 which is divisible by 9.
Here, a = 108, d = 117 – 108 = 9 and an = l = 198
`\implies` 198 = 108 + (n – 1)9 ...[∵ an = l = a + (n – 1)d]
`\implies` 90 = (n – 1)9
`\implies` n – 1 = 10
`\implies` n = 11
∴ Sum of terms between 100 and 200 which is divisible by 9 is
Sn = `n/2[2a + (n - 1)d]`
`\implies` S11 = `11/2[2(108) + (11 - 1)9]`
= `11/2[216 + 90]`
= `11/2 xx 306`
= 11 × 153
= 1683
Hence, required sum of the integers between 100 and 200 that are divisible by 9 is 1683.
ii. The sum of the integers between 100 and 200 which is not divisible by 9 = (sum of total numbers between 100 and 200) – (sum of total numbers between 100 and 200 which is divisible by 9) ...(i)
Total numbers between 100 and 200 is 101, 102, 103,..., 199
Here, a = 101, d = 102 – 101 = 1 and an = l = 199
`\implies` 199 = 101 + (n – 1)1 ...[∵ an = l = a + (n – 1)d]
`\implies` (n – 1) = 98
`\implies` n = 99
Sum of terms between 100 and 200,
Sn = `n/2[2a + (n - 1)d]`
`\implies` S99 = `99/2[2(101) + (99 - 1)1]`
= `99/2[202 + 98]`
= `99/2 xx 300`
= 99 × 150
= 14850
From equation (i), sum of the integers between 100 and 200 which is not divisible by 9
= 14850 – 1683 ...[From part (i)]
= 13167
Hence, the required sum is 13167.
RELATED QUESTIONS
Ramkali required Rs 2,500 after 12 weeks to send her daughter to school. She saved Rs 100 in the first week and increased her weekly saving by Rs 20 every week. Find whether she will be able to send her daughter to school after 12 weeks.
What value is generated in the above situation?
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.
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)
If the sum of first m terms of an A.P. is the same as the sum of its first n terms, show that the sum of its first (m + n) terms is zero
If (m + 1)th term of an A.P is twice the (n + 1)th term, prove that (3m + 1)th term is twice the (m + n + 1)th term.
Find the sum of the following arithmetic progressions:
`(x - y)/(x + y),(3x - 2y)/(x + y), (5x - 3y)/(x + y)`, .....to n terms
Is 184 a term of the AP 3, 7, 11, 15, ….?
The sum of the first n terms of an AP is given by `s_n = ( 3n^2 - n) ` Find its
(i) nth term,
(ii) first term and
(iii) common difference.
Kargil’s temperature was recorded in a week from Monday to Saturday. All readings were in A.P. The sum of temperatures of Monday and Saturday was 5°C more than sum of temperatures of Tuesday and Saturday. If temperature of Wednesday was –30° celsius then find the temperature on the other five days.
Choose the correct alternative answer for the following question .
What is the sum of the first 30 natural numbers ?
For an given A.P., t7 = 4, d = −4, then a = ______.
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.
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 ?
Q.18
The sum of the first three numbers in an Arithmetic Progression is 18. If the product of the first and the third term is 5 times the common difference, find the three numbers.
The sum of first six terms of an arithmetic progression is 42. The ratio of the 10th term to the 30th term is `(1)/(3)`. Calculate the first and the thirteenth term.
In an A.P., if Sn = 3n2 + 5n and ak = 164, find the value of k.
Calculate the sum of 35 terms in an AP, whose fourth term is 16 and ninth term is 31.
Find the sum of first 20 terms of an A.P. whose nth term is given as an = 5 – 2n.
The sum of 40 terms of the A.P. 7 + 10 + 13 + 16 + .......... is ______.
