Advertisements
Advertisements
प्रश्न
Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.
Advertisements
उत्तर
The odd integers between 1 and 1000 that are divisible by 3 are:
3, 9, 15, 21...999
Here, we have:
\[a = 3, d = 6\]
\[ a_n = 999\]
\[ \Rightarrow 3 + (n - 1)6 = 999\]
\[ \Rightarrow 3 + 6n - 6 = 999\]
\[ \Rightarrow 6n = 1002\]
\[ \Rightarrow n = 167\]
\[ S_n = \frac{n}{2}\left[ 2a + (n - 1)d \right]\]
\[ \Rightarrow S_{167} = \frac{167}{2}\left[ 2 \times 3 + (167 - 1)6 \right]\]
\[ \Rightarrow S_{167} = \frac{167}{2}\left[ 1002 \right] = 83667\]
\[\text { Hence, proved } .\]
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
If the sum of n terms of an A.P. is 3n2 + 5n and its mth term is 164, find the value of m.
if `(a^n + b^n)/(a^(n-1) + b^(n-1))` is the A.M. between a and b, then find the value of n.
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
The nth term of a sequence is given by an = 2n + 7. Show that it is an A.P. Also, find its 7th term.
Find:
nth term of the A.P. 13, 8, 3, −2, ...
Which term of the A.P. 3, 8, 13, ... is 248?
How many terms are there in the A.P. 7, 10, 13, ... 43 ?
The 4th term of an A.P. is three times the first and the 7th term exceeds twice the third term by 1. Find the first term and the common difference.
Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.
The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.
The angles of a quadrilateral are in A.P. whose common difference is 10°. Find the angles.
Find the sum of the following arithmetic progression :
3, 9/2, 6, 15/2, ... to 25 terms
Find the sum of first n natural numbers.
Find the sum of n terms of the A.P. whose kth terms is 5k + 1.
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
If \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:
\[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P.
If a2, b2, c2 are in A.P., prove that \[\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}\] are in A.P.
If a, b, c is in A.P., prove that:
(a − c)2 = 4 (a − b) (b − c)
If a, b, c is in A.P., prove that:
a2 + c2 + 4ac = 2 (ab + bc + ca)
If x, y, z are in A.P. and A1 is the A.M. of x and y and A2 is the A.M. of y and z, then prove that the A.M. of A1 and A2 is y.
A man is employed to count Rs 10710. He counts at the rate of Rs 180 per minute for half an hour. After this he counts at the rate of Rs 3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.
If the sum of n terms of an AP is 2n2 + 3n, then write its nth term.
Write the sum of first n odd natural numbers.
If \[\frac{3 + 5 + 7 + . . . + \text { upto n terms }}{5 + 8 + 11 + . . . . \text { upto 10 terms }}\] 7, then find the value of n.
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is
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
In the arithmetic progression whose common difference is non-zero, the sum of first 3 n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2 n terms to the next 2 nterms is
If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are
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 \[\frac{S_1}{S_2}\] =
If in an A.P., Sn = n2p and Sm = m2p, where Sr denotes the sum of r terms of the A.P., then Sp is equal to
Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively.
The first three of four given numbers are in G.P. and their last three are in A.P. with common difference 6. If first and fourth numbers are equal, then the first number is
If there are (2n + 1) terms in an A.P., then prove that the ratio of the sum of odd terms and the sum of even terms is (n + 1) : n
If 9 times the 9th term of an A.P. is equal to 13 times the 13th term, then the 22nd term of the A.P. is ______.
If in an A.P., Sn = qn2 and Sm = qm2, where Sr denotes the sum of r terms of the A.P., then Sq equals ______.
The sum of terms equidistant from the beginning and end in an A.P. is equal to ______.
If the ratio of the sum of n terms of two APs is 2n:(n + 1), then the ratio of their 8th terms is ______.
