हिंदी

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 - Mathematics

Advertisements
Advertisements

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`

योग
Advertisements

उत्तर

It is known that the general term of an A.P. is an = a + (n – 1)d

∴ According to the given information,

shaalaa.com
  क्या इस प्रश्न या उत्तर में कोई त्रुटि है?

वीडियो ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्न

Sum of the first p, q and r terms of an A.P. are a, b and c, respectively.

Prove that `a/p (q - r) + b/q (r- p) + c/r (p - q) = 0`


The ratio of the sums of m and n terms of an A.P. is m2n2. Show that the ratio of mth and nthterm is (2m – 1): (2n – 1)


The difference between any two consecutive interior angles of a polygon is 5°. If the smallest angle is 120°, find the number of the sides of the polygon.


A manufacturer reckons that the value of a machine, which costs him Rs 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years.


Let < an > be a sequence. Write the first five term in the following:

a1 = a2 = 2, an = a− 1 − 1, n > 2


The Fibonacci sequence is defined by a1 = 1 = a2, an = an − 1 + an − 2 for n > 2

Find `(""^an +1)/(""^an")` for n = 1, 2, 3, 4, 5.

 


Find:

nth term of the A.P. 13, 8, 3, −2, ...


If the sequence < an > is an A.P., show that am +n +am − n = 2am.


Which term of the A.P. 3, 8, 13, ... is 248?


Which term of the sequence 24, \[23\frac{1}{4,} 22\frac{1}{2,} 21\frac{3}{4}\]....... is the first negative term?


Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely real ?


How many numbers of two digit are divisible by 3?


How many numbers are there between 1 and 1000 which when divided by 7 leave remainder 4?


The sum of three terms of an A.P. is 21 and the product of the first and the third terms exceeds the second term by 6, find three terms.


The sum of three numbers in A.P. is 12 and the sum of their cubes is 288. Find the numbers.


If the sum of three numbers in A.P. is 24 and their product is 440, find the numbers.


Find the sum of the following arithmetic progression :

41, 36, 31, ... to 12 terms


Find the sum of all integers between 100 and 550, which are divisible by 9.


Find the sum of the series:
3 + 5 + 7 + 6 + 9 + 12 + 9 + 13 + 17 + ... to 3n terms.


Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.


Solve: 

25 + 22 + 19 + 16 + ... + x = 115


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 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 \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:

\[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] 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.


If a, b, c is in A.P., prove that:

 (a − c)2 = 4 (a − b) (b − c)


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.


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


Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively. 


If for an arithmetic progression, 9 times nineth term is equal to 13 times thirteenth term, then value of twenty second term is ____________.


If the sum of m terms of an A.P. is equal to the sum of either the next n terms or the next p terms, then prove that `(m + n) (1/m - 1/p) = (m + p) (1/m - 1/n)`


The first term of an A.P.is a, and the sum of the first p terms is zero, show that the sum of its next q terms is `(-a(p + q)q)/(p - 1)`


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 ______.


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 ______.


The number of terms in an A.P. is even; the sum of the odd terms in lt is 24 and that the even terms is 30. If the last term exceeds the first term by `10 1/2`, then the number of terms in the A.P. is ______.


If b2, a2, c2 are in A.P., then `1/(a + b), 1/(b + c), 1/(c + a)` will be in ______


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×