Advertisements
Advertisements
प्रश्न
If first term of an A.P. is a, second term is b and last term is c, then show that sum of all terms is \[\frac{\left( a + c \right) \left( b + c - 2a \right)}{2\left( b - a \right)}\].
Advertisements
उत्तर
a, b, ..., c
t1 = a, d = b - a, tn = c
We know that
tn = a + (n - 1)d
c = a + (n - 1)(b - a)
`(c-a)/(b-a)=n-1`
`(c-a)/(b-a)+1/1=n`
`(c-a+b-a)/(b-a)=n`
∴ n = `(c+b-2a)/(b-a)` ...(1)
Now,
Sn = `n/2[t_1 + t_n]`
Sn = `(c+b-2a)/((b-a)2)[a+c]`
Sn = `((a+c)(b+c-2a))/(2(b-a))`
Hence proved.
APPEARS IN
संबंधित प्रश्न
If the ratio of the sum of first n terms of two A.P’s is (7n +1): (4n + 27), find the ratio of their mth terms.
Find the sum of first 8 multiples of 3
Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.
Find the middle term of the AP 6, 13, 20, …., 216.
In a flower bed, there are 43 rose plants in the first row, 41 in second, 39 in the third, and so on. There are 11 rose plants in the last row. How many rows are there in the flower bed?
If the numbers (2n – 1), (3n+2) and (6n -1) are in AP, find the value of n and the numbers
If `4/5 `, a, 2 are in AP, find the value of a.
Write an A.P. whose first term is a and the common difference is d in the following.
a = 10, d = 5
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`
In an A.P. the 10th term is 46 sum of the 5th and 7th term is 52. Find the A.P.
There are 37 terms in an A.P., the sum of three terms placed exactly at the middle is 225 and the sum of last three terms is 429. Write the A.P.
If `4/5` , a, 2 are three consecutive terms of an A.P., then find the value of a.
If the first, second and last term of an A.P. are a, b and 2a respectively, its sum is
In an AP. Sp = q, Sq = p and Sr denotes the sum of first r terms. Then, Sp+q is equal to
Q.4
Q.15
How many terms of the A.P. 24, 21, 18, … must be taken so that the sum is 78? Explain the double answer.
If the third term of an A.P. is 1 and 6th term is – 11, find the sum of its first 32 terms.
What is the sum of an odd numbers between 1 to 50?
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?
