Advertisements
Advertisements
Question
The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.
Advertisements
Solution
We know that,
an = a + (n − 1) d
a3 = a + (3 − 1) d
a3 = a + 2d
Similarly, a7 = a + 6d
Given that, a3 + a7 = 6
(a + 2d) + (a + 6d) = 6
2a + 8d = 6
a + 4d = 3
a = 3 − 4d ...(i)
Also, it is given that (a3) × (a7) = 8
⇒ (a + 2d) × (a + 6d) = 8
From equation (i),
(3 - 4d + 2d) × (3 - 4d + 6d) = 8
(3 - 2d) × (3 + 2d) = 8
9 - 4d2 = 8
4d2 = 9 - 8
4d2 = 1
d2 = `1/4`
d = `± 1/2`
d = `1/2 or 1/2`
From the equation (i)
`("When" "d" 1/2)`
a = 3 - 4d
a = `3 -4(1/4)`
a = 3 - 2
a = 1
When d is − `1/2`
a = `3 - 4(-1/2)`
a = 3 + 2
a = 5
`S_n = n/2 [2a(n-1)s]`
(When a is 1 and d is `1/2`)
`S_16 = (16/2)[2"a" + (16 - 1)"d"]`
`S_16 = 8[2 xx 1 + 15(1/2)]`
`S_16 = 8(2 + 15/2)`
`S_16 = 8 xx 19/2`
S16 = 76
(When a is 5 and d is `-1/2`)
a = `3 - 4(-1/2)`
a = `3 + 4/2`
a = 5
S16 = `(16/2)[2a + (n - 1)d]`
= 8`[2(5) + 15(-1/2)]`
= `8[10 - 15/2]`
= `8 xx 5/2`
= 20
RELATED QUESTIONS
If Sn1 denotes the sum of first n terms of an A.P., prove that S12 = 3(S8 − S4).
Show that a1, a2,..., an... form an AP where an is defined as below:
an = 9 − 5n
Also, find the sum of the first 15 terms.
Find the sum 25 + 28 + 31 + ….. + 100
Is -150 a term of the AP 11, 8, 5, 2, ……?
If the pth term of an AP is q and its qth term is p then show that its (p + q)th term is zero
The first three terms of an AP are respectively (3y – 1), (3y + 5) and (5y + 1), find the value of y .
If the numbers a, 9, b, 25 from an AP, find a and b.
Write the next term for the AP` sqrt( 8), sqrt(18), sqrt(32),.........`
How many terms of the A.P. 21, 18, 15, … must be added to get the sum 0?
Find the sum of all natural numbers between 200 and 400 which are divisible by 7.
Find the sum of all multiples of 9 lying between 300 and 700.
If the first term of an A.P. is 2 and common difference is 4, then the sum of its 40 terms is
The given terms are 2k + 1, 3k + 3 and 5k − 1. find AP.
Find the sum of all members from 50 to 250 which divisible by 6 and find t13.
How many terms of the A.P. 27, 24, 21, …, should be taken so that their sum is zero?
Find the sum of first 1000 positive integers.
Activity :- Let 1 + 2 + 3 + ........ + 1000
Using formula for the sum of first n terms of an A.P.,
Sn = `square`
S1000 = `square/2 (1 + 1000)`
= 500 × 1001
= `square`
Therefore, Sum of the first 1000 positive integer is `square`
The sum of first n terms of the series a, 3a, 5a, …….. is ______.
The ratio of the 11th term to the 18th term of an AP is 2 : 3. Find the ratio of the 5th term to the 21st term, and also the ratio of the sum of the first five terms to the sum of the first 21 terms.
Sum of 1 to n natural number is 45, then find the value of n.
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?
