Advertisements
Advertisements
प्रश्न
Find the sum of all natural numbers between 1 and 100, which are divisible by 2 or 5.
Advertisements
उत्तर
We have to find the sum of all the natural numbers that are divisible by 2 or 5
Required Sum = Sum of the natural numbers between 1 and 100 that are divisible by 2 + Sum of the natural numbers between 1 and 100 that are divisible by 5
− Sum of the natural numbers between 1 and 100 that are divisible by 2 and 5, i.e by 10
\[= \left( 2 + 4 + 6 + 8 + . . . + 98 \right) + \left( 5 + 10 + 15 + . . . + 95 \right) - \left( 10 + 20 + 30 + . . . + 90 \right)\]
\[ = \frac{50}{2}\left( 2 + 98 \right) + \frac{20}{2}\left( 5 + 95 \right) - \frac{10}{2}\left( 10 + 90 \right)\]
\[ = 2500 + 1000 - 500\]
\[ = 3000\]
APPEARS IN
संबंधित प्रश्न
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4: 9n + 6. Find the ratio of their 18th terms
Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and (m – 1)th numbers is 5:9. Find the value of m.
Find the sum of all numbers between 200 and 400 which are divisible by 7.
The pth, qth and rth terms of an A.P. are a, b, c respectively. Show that (q – r )a + (r – p )b + (p – q )c = 0
Let < an > be a sequence. Write the first five term in the following:
a1 = 1 = a2, an = an − 1 + an − 2, n > 2
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, 7\sqrt{2}, . . .\]
The nth term of a sequence is given by an = 2n2 + n + 1. Show that it is not an A.P.
Find:
18th term of the A.P.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2},\]
If the sequence < an > is an A.P., show that am +n +am − n = 2am.
Which term of the A.P. 84, 80, 76, ... is 0?
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely real ?
Find the 12th term from the following arithmetic progression:
3, 5, 7, 9, ... 201
The sum of three numbers in A.P. is 12 and the sum of their cubes is 288. Find the numbers.
Find the sum of the following arithmetic progression :
(x − y)2, (x2 + y2), (x + y)2, ... to n terms
Find the sum of the following serie:
(a − b)2 + (a2 + b2) + (a + b)2 + ... + [(a + b)2 + 6ab]
Find the r th term of an A.P., the sum of whose first n terms is 3n2 + 2n.
The first term of an A.P. is 2 and the last term is 50. The sum of all these terms is 442. Find the common difference.
If the 5th and 12th terms of an A.P. are 30 and 65 respectively, what is the sum of first 20 terms?
If the sum of n terms of an A.P. is nP + \[\frac{1}{2}\] n (n − 1) Q, where P and Q are constants, find the common difference.
The sums of first n terms of two A.P.'s are in the ratio (7n + 2) : (n + 4). Find the ratio of their 5th terms.
If a, b, c is in A.P., then show that:
b + c − a, c + a − b, a + b − c are in A.P.
If a, b, c is in A.P., prove that:
a2 + c2 + 4ac = 2 (ab + bc + ca)
Write the sum of first n odd natural numbers.
Write the value of n for which n th terms of the A.P.s 3, 10, 17, ... and 63, 65, 67, .... are equal.
If m th term of an A.P. is n and nth term is m, then write its pth term.
In n A.M.'s are introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3 : 1, then the value of n is
If, S1 is the sum of an arithmetic progression of 'n' odd number of terms and S2 the sum of the terms of the series in odd places, then \[\frac{S_1}{S_2}\] =
Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively.
If a1, a2, ..., an are in A.P. with common difference d (where d ≠ 0); then the sum of the series sin d (cosec a1 cosec a2 + cosec a2 cosec a3 + ...+ cosec an–1 cosec an) is equal to cot a1 – cot an
A man saved Rs 66000 in 20 years. In each succeeding year after the first year he saved Rs 200 more than what he saved in the previous year. How much did he save in the first year?
A man accepts a position with an initial salary of Rs 5200 per month. It is understood that he will receive an automatic increase of Rs 320 in the very next month and each month thereafter. Find his salary for the tenth month
If the sum of p terms of an A.P. is q and the sum of q terms is p, show that the sum of p + q terms is – (p + q). Also, find the sum of first p – q terms (p > q).
If the sum of n terms of a sequence is quadratic expression then it always represents an A.P
Let 3, 6, 9, 12 ....... upto 78 terms and 5, 9, 13, 17 ...... upto 59 be two series. Then, the sum of the terms common to both the series is equal to ______.
The sum of n terms of an AP is 3n2 + 5n. The number of term which equals 164 is ______.
If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is ______.
If b2, a2, c2 are in A.P., then `1/(a + b), 1/(b + c), 1/(c + a)` will be in ______
