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).
Advertisements
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
If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term
A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual installments of Rs 500 plus 12% interest on the unpaid amount. How much will be the tractor cost him?
A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually. Find the amount in 15th year since he deposited the amount and also calculate the total amount after 20 years.
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.
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.
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
−1, 1/4, 3/2, 11/4, ...
Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, 7\sqrt{2}, . . .\]
Find:
18th term of the A.P.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2},\]
Is 302 a term of the A.P. 3, 8, 13, ...?
The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.
Find the sum of the following arithmetic progression :
1, 3, 5, 7, ... to 12 terms
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:
101 + 99 + 97 + ... + 47
Find the sum of first n odd natural numbers.
Find the sum of all odd numbers between 100 and 200.
Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.
Find the sum of n terms of the A.P. whose kth terms is 5k + 1.
How many terms of the A.P. −6, \[- \frac{11}{2}\], −5, ... are needed to give the sum −25?
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.
A man saves Rs 32 during the first year. Rs 36 in the second year and in this way he increases his savings by Rs 4 every year. Find in what time his saving will be Rs 200.
A piece of equipment cost a certain factory Rs 600,000. If it depreciates in value, 15% the first, 13.5% the next year, 12% the third year, and so on. What will be its value at the end of 10 years, all percentages applying to the original cost?
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. whose nth term is xn + y.
Write the common difference of an A.P. the sum of whose first n terms is
If m th term of an A.P. is n and nth term is m, then write its pth term.
If the sums of n terms of two AP.'s are in the ratio (3n + 2) : (2n + 3), then find the ratio of their 12th terms.
If the sum of n terms of an A.P. be 3 n2 − n and its common difference is 6, then its first term 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 \[\frac{S_1}{S_2}\] =
Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively.
If a, b, c are in A.P. and x, y, z are in G.P., then the value of xb − c yc − a za − b is
If for an arithmetic progression, 9 times nineth term is equal to 13 times thirteenth term, then value of twenty second term 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
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.
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)`
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 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 the sum of n terms of a sequence is quadratic expression then it always represents an A.P
