Advertisements
Advertisements
प्रश्न
Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.
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
In an A.P., if pth term is 1/q and qth term is 1/p, prove that the sum of first pq terms is 1/2 (pq + 1) where `p != q`
A man starts repaying a loan as first installment of Rs. 100. If he increases the installment by Rs 5 every month, what amount he will pay in the 30th installment?
Find the sum of integers from 1 to 100 that are divisible by 2 or 5.
If the nth term an of a sequence is given by an = n2 − n + 1, write down its first five terms.
A sequence is defined by an = n3 − 6n2 + 11n − 6, n ϵ N. Show that the first three terms of the sequence are zero and all other terms are positive.
The nth term of a sequence is given by an = 2n2 + n + 1. Show that it is not an A.P.
Is 302 a term of the A.P. 3, 8, 13, ...?
The first term of an A.P. is 5, the common difference is 3 and the last term is 80; find the number of terms.
The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.
If the nth term of the A.P. 9, 7, 5, ... is same as the nth term of the A.P. 15, 12, 9, ... find n.
Find the 12th term from the following arithmetic progression:
3, 5, 7, 9, ... 201
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.
If < an > is an A.P. such that \[\frac{a_4}{a_7} = \frac{2}{3}, \text { find }\frac{a_6}{a_8}\].
Find the sum of the following arithmetic progression :
1, 3, 5, 7, ... to 12 terms
Find the sum of the following arithmetic progression :
3, 9/2, 6, 15/2, ... to 25 terms
Find the sum of the following serie:
2 + 5 + 8 + ... + 182
Find the sum of all integers between 50 and 500 which are divisible by 7.
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
How many terms of the A.P. −6, \[- \frac{11}{2}\], −5, ... are needed to give the sum −25?
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 \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
bc, ca, ab are in A.P.
Show that x2 + xy + y2, z2 + zx + x2 and y2 + yz + z2 are consecutive terms of an A.P., if x, y and z are in A.P.
A man saved Rs 16500 in ten years. In each year after the first he saved Rs 100 more than he did in the receding year. How much did he save in the first year?
Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual instalments of Rs 1000 plus 10% interest on the unpaid amount. How much the scooter will cost him.
Write the common difference of an A.P. the sum of whose first n terms is
If log 2, log (2x − 1) and log (2x + 3) are in A.P., write the value of x.
Write the sum of first n odd natural numbers.
Sum of all two digit numbers which when divided by 4 yield unity as remainder 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
The first and last terms of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be
Mark the correct alternative in the following question:
The 10th common term between the A.P.s 3, 7, 11, 15, ... and 1, 6, 11, 16, ... is
Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively.
The first term of an A.P. is a, the second term is b and the last term is c. Show that the sum of the A.P. is `((b + c - 2a)(c + a))/(2(b - a))`.
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 ______.
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 a1, a2, a3, .......... are an A.P. such that a1 + a5 + a10 + a15 + a20 + a24 = 225, then a1 + a2 + a3 + ...... + a23 + a24 is equal to ______.
