Advertisements
Advertisements
प्रश्न
If S1 be the sum of (2n + 1) terms of an A.P. and S2 be the sum of its odd terms, then prove that: S1 : S2 = (2n + 1) : (n + 1).
उत्तर
To prove: S1 : S2 = (2n + 1) : (n + 1)
We know that the sum of AP is given by the formula:
`S = n/2(2a + (n - 1)d)`
Substituting the values in the above equation,
`S_1 = (2n + 1)/2 (2a + 2nd)`
For the sum of odd terms, it is given by,
`S_2 = a_1 + a_3 + a_5 + .....a_(2n) + 1`
`S_2 = a + a + 2d + a + 4d + .... + a + 2nd`
`S_2 = (n + 1)a + n (n + 1)d`
`S_2 = (n + 1)(a + nd)`
Hence,
`S_1 : S_2 = (2n + 1)/(n + 1)`
APPEARS IN
संबंधित प्रश्न
Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.
If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term
If the sum of n terms of an A.P. is (pn + qn2), where p and q are constants, find the common difference.
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4: 9n + 6. Find the ratio of their 18th terms
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`
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.
Let the sum of n, 2n, 3n terms of an A.P. be S1, S2 and S3, respectively, show that S3 = 3 (S2– S1)
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
3, −1, −5, −9 ...
If the sequence < an > is an A.P., show that am +n +am − n = 2am.
Which term of the sequence 24, \[23\frac{1}{4,} 22\frac{1}{2,} 21\frac{3}{4}\]....... is the first negative term?
The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.
Find the 12th term from the following arithmetic progression:
3, 5, 7, 9, ... 201
Find the sum of the following arithmetic progression :
41, 36, 31, ... to 12 terms
Find the sum of the following serie:
(a − b)2 + (a2 + b2) + (a + b)2 + ... + [(a + b)2 + 6ab]
Find the sum of first n odd natural numbers.
Find the sum of all even integers between 101 and 999.
The number of terms of an A.P. is even; the sum of odd terms is 24, of the even terms is 30, and the last term exceeds the first by \[10 \frac{1}{2}\] ,find the number of terms and the series.
If 12th term of an A.P. is −13 and the sum of the first four terms is 24, what is the sum of first 10 terms?
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
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 \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:
a (b +c), b (c + a), c (a +b) 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:
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 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is
Sum of all two digit numbers which when divided by 4 yield unity as remainder is
If Sn denotes the sum of first n terms of an A.P. < an > such that
If the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?
If second, third and sixth terms of an A.P. are consecutive terms of a G.P., write the common ratio of the G.P.
Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively.
The pth term of an A.P. is a and qth term is b. Prove that the sum of its (p + q) terms is `(p + q)/2[a + b + (a - b)/(p - q)]`.
Show that (x2 + xy + y2), (z2 + xz + x2) and (y2 + yz + z2) are consecutive terms of an A.P., if x, y and z are in A.P.
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to ______.
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 internal angles of a convex polygon are in A.P. The smallest angle is 120° and the common difference is 5°. The number to sides of the polygon is ______.
