Advertisements
Advertisements
प्रश्न
Show that the sum of an AP whose first term is a, the second term b and the last term c, is equal to `((a + c)(b + c - 2a))/(2(b - a))`
Advertisements
उत्तर
Given that, the AP is a, b, c
Here, first term = a,
Common difference = b – a
And last term, l = an = c
∵ an = l = a + (n – 1)d
⇒ c = a + (n – 1)(b – a)
⇒ (n – 1) = `(c - a)/(b - a)`
n = `(c - a)/(b - a) + 1`
⇒ n = `(c - a + b - a)/(b - a)`
= `(c + b - 2a)/(b - a)` ...(i)
∴ Sum of an AP,
Sn = `n/2 [2a + (n - 1)d]`
= `((b + c - 2a))/(2(b - a)) [2a + {(b + c - 2a)/(b - a) - 1}(b - a)]`
= `((b + c - 2a))/(2(b - a))[2a + (c - a)/(b - a) * (b - a)]`
= `((b + c - 2a))/(2(b - a)) (2a + c - a)`
= `((b + c - 2a))/(2(b - a)) * (a + c)`
Hence proved.
APPEARS IN
संबंधित प्रश्न
Find four numbers in A.P. whose sum is 20 and the sum of whose squares is 120
Find the sum of first 20 terms of the following A.P. : 1, 4, 7, 10, ........
A small terrace at a football field comprises 15 steps, each of which is 50 m long and built of solid concrete. Each step has a rise of `1/4` m and a tread of `1/2` m (See figure). Calculate the total volume of concrete required to build the terrace.
[Hint: Volume of concrete required to build the first step = `1/4 xx 1/2 xx 50 m^3`]
If the sum of first m terms of an A.P. is the same as the sum of its first n terms, show that the sum of its first (m + n) terms is zero
Find the sum of all even integers between 101 and 999.
Find the sum of the first 13 terms of the A.P: -6, 0, 6, 12,....
Find the 8th term from the end of the AP 7, 10, 13, ……, 184.
Find the value of x for which the numbers (5x + 2), (4x - 1) and (x + 2) are in AP.
Find the three numbers in AP whose sum is 15 and product is 80.
Find the first term and common difference for the A.P.
127, 135, 143, 151,...
First term and the common differences of an A.P. are 6 and 3 respectively; find S27.
Solution: First term = a = 6, common difference = d = 3, S27 = ?
Sn = `"n"/2 [square + ("n" - 1)"d"]` - Formula
Sn = `27/2 [12 + (27 - 1)square]`
= `27/2 xx square`
= 27 × 45
S27 = `square`
Find the sum: 1 + 3 + 5 + 7 + ... + 199 .
A man is employed to count Rs 10710. He counts at the rate of Rs 180 per minute for half an hour. After this he counts at the rate of Rs 3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.
Write the nth term of the \[A . P . \frac{1}{m}, \frac{1 + m}{m}, \frac{1 + 2m}{m}, . . . .\]
If the nth term of an A.P. is 2n + 1, then the sum of first n terms of the A.P. is
For an A.P., If t1 = 1 and tn = 149 then find Sn.
Activitry :- Here t1= 1, tn = 149, Sn = ?
Sn = `"n"/2 (square + square)`
= `"n"/2 xx square`
= `square` n, where n = 75
The sum of all odd integers between 2 and 100 divisible by 3 is ______.
Find the middle term of the AP. 95, 86, 77, ........, – 247.
In the month of April to June 2022, the exports of passenger cars from India increased by 26% in the corresponding quarter of 2021-22, as per a report. A car manufacturing company planned to produce 1800 cars in 4th year and 2600 cars in 8th year. Assuming that the production increases uniformly by a fixed number every year. |
Based on the above information answer the following questions.
- Find the production in the 1st year
- Find the production in the 12th year.
- Find the total production in first 10 years.
[OR]
In how many years will the total production reach 31200 cars?
Solve the equation:
– 4 + (–1) + 2 + 5 + ... + x = 437
