Advertisements
Advertisements
प्रश्न
The sum of third and seventh term of an A. P. is 6 and their product is 8. Find the first term and the common difference of the A. P.
उत्तर
`t_n=a(n-1)d`
∴ `t_3=a+(3-1)d=a+2d`
`t_7=a+(3-1)d=a+2d`
∴ `t_3+t_7=(a+2d)+(a+6d)=2a+8d`
∴ `2a+8d=6`
∴ a+4d=3 ................(I)
`t_3xxt_7=(a+2d)(a+6d)`
= `(a+4d-2d) (a+4d+2d)`
=`(3-2d) (3+2d) `......................from (I)
∴ `(3-2d)(3+2d)=8`
∴ `9-4d^2=8`
∴` 4d^2=1 d^2=1/4 d=1/2 or d=-1/2`
Now, `if d=1/2`
`a+4xx1/2=3`................ from (I)
`a=1 `
If ` d=-1/2`
`a+4xx(-1/2)=3.......` from (I)
`a=5`
∴ the first term of the A. P. is 1 and the common difference is `1/2.`
or , the first term of the A. P. is 5 and the common difference is `-1/2`.
APPEARS IN
संबंधित प्रश्न
The houses in a row numbered consecutively from 1 to 49. Show that there exists a value of x such that sum of numbers of houses preceding the house numbered x is equal to sum of the numbers of houses following x.
If the mth term of an A.P. is 1/n and the nth term is 1/m, show that the sum of mn terms is (mn + 1)
If the term of m terms of an A.P. is the same as the sum of its n terms, show that the sum of its (m + n) terms is zero
Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.
In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato and other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line.
A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?
[Hint: to pick up the first potato and the second potato, the total distance (in metres) run by a competitor is 2 × 5 + 2 ×(5 + 3)]
A sum of ₹2800 is to be used to award four prizes. If each prize after the first is ₹200 less than the preceding prize, find the value of each of the prizes
The sum of three numbers in AP is 3 and their product is -35. Find the numbers.
How many three-digit natural numbers are divisible by 9?
Which term of the AP 21, 18, 15, … is zero?
Find four consecutive terms in an A.P. whose sum is 12 and sum of 3rd and 4th term is 14.
(Assume the four consecutive terms in A.P. are a – d, a, a + d, a +2d)
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 \[\frac{\left( a + c \right) \left( b + c - 2a \right)}{2\left( b - a \right)}\].
For what value of n, the nth terms of the arithmetic progressions 63, 65, 67, ... and 3, 10, 17, ... equal?
The sum of first 9 terms of an A.P. is 162. The ratio of its 6th term to its 13th term is 1 : 2. Find the first and 15th term of the A.P.
If the sum of three consecutive terms of an increasing A.P. is 51 and the product of the first and third of these terms is 273, then the third 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 the sum of the first four terms of an AP is 40 and that of the first 14 terms is 280. Find the sum of its first n terms.
Find the sum:
1 + (–2) + (–5) + (–8) + ... + (–236)
Find the sum:
`4 - 1/n + 4 - 2/n + 4 - 3/n + ...` upto n terms
The sum of A.P. 4, 7, 10, 13, ........ upto 20 terms is ______.
