Advertisements
Advertisements
प्रश्न
If the first term of an A.P. is p, second term is q and last term is r, then show that sum of all terms is `(q + r - 2p) xx ((p + r))/(2(q - p))`.
Advertisements
उत्तर
Given: First term (t1) = a = p, Second term (t2) = q, tn = r
Common difference (d) = t2 – t1 = q – p
According to the question,
tn = a + (n – 1) × d
r = p + (n – 1) × (q – p)
(r – p) = (n – 1) × (q – p)
n – 1 = `(r - p)/(q - p)`
n = `(r - p)/(q - p) + 1`
n = `(r - p + q - p)/(q - p)`
n = `(r + q - 2p)/(q - p)`
We know
Sn = `n/2[2a + (n - 1)d]`
= `(r + q - 2p)/(2(q - p)) [2p + ((r + q - 2p)/(q - p) - 1) (q - p)]`
= `(r + q - 2p)/(2(q - p)) [2p + ((r + q - 2p - (q - p))/(q - p)) (q - p)]`
= `(r + q - 2p)/(2(q - p)) [((r + q - 2p - q + p)/(q - p)) (q - p)]`
= `(r + q - 2p)/(2(q - p))[2p + ((r - p)/(q - p)) (q - p)]`
= `(r + q - 2p)/(2(q - p))[2p + r - p]`
= `(r + q - 2p)/(2(q - p))[r + p]`
Sn = `(q + r - 2p) xx ((p + r))/(2(q - p))`
Hence proved.
APPEARS IN
संबंधित प्रश्न
Find the sum of the first 15 terms of each of the following sequences having the nth term as
bn = 5 + 2n
In an A.P. the first term is 25, nth term is –17 and the sum of n terms is 132. Find n and the common difference.
If the 8th term of an A.P. is 37 and the 15th term is 15 more than the 12th term, find the A.P. Also, find the sum of first 20 terms of A.P.
Write the next term of the AP `sqrt(8), sqrt(18), sqrt(32),`....
Find an AP whose 4th term is 9 and the sum of its 6th and 13th terms is 40.
The 9th term of an A.P. is equal to 6 times its second term. If its 5th term is 22, find the A.P.
The sum of 5th and 9th terms of an A.P. is 30. If its 25th term is three times its 8th term, find the A.P.
Find the sum: 1 + 3 + 5 + 7 + ... + 199 .
A sum of Rs. 700 is to be paid to give seven cash prizes to the students of a school for their overall academic performance. If the cost of each prize is Rs. 20 less than its preceding prize; find the value of each of the prizes.
Q.17
How many terms of the series 18 + 15 + 12 + ........ when added together will give 45?
If the second term and the fourth term of an A.P. are 12 and 20 respectively, then find the sum of first 25 terms:
Find t21, if S41 = 4510 in an A.P.
The sum of first ten natural number is ______.
If the first term of an AP is –5 and the common difference is 2, then the sum of the first 6 terms is ______.
Find the sum of all the 11 terms of an A.P. whose middle most term is 30.
Find the sum of those integers from 1 to 500 which are multiples of 2 as well as of 5.
Jaspal Singh repays his total loan of Rs. 118000 by paying every month starting with the first instalment of Rs. 1000. If he increases the instalment by Rs. 100 every month, what amount will be paid by him in the 30th instalment? What amount of loan does he still have to pay after the 30th instalment?
The students of a school decided to beautify the school on the Annual Day by fixing colourful flags on the straight passage of the school. They have 27 flags to be fixed at intervals of every 2 m. The flags are stored at the position of the middle most flag. Ruchi was given the responsibility of placing the flags. Ruchi kept her books where the flags were stored. She could carry only one flag at a time. How much distance did she cover in completing this job and returning back to collect her books? What is the maximum distance she travelled carrying a flag?
The nth term of an Arithmetic Progression (A.P.) is given by the relation Tn = 6(7 – n)..
Find:
- its first term and common difference
- sum of its first 25 terms
