Advertisements
Advertisements
प्रश्न
The sum of three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find a, b, c.
Advertisements
उत्तर
Let the first term of the A.P. be a and the common difference be d.
∴ a = a , b = a + d and c = a + 2d
\[a + b + c = 18\]
\[ \Rightarrow a + \left( a + d \right) + \left( a + 2d \right) = 18\]
\[ \Rightarrow 3a + 3d = 18 \]
\[ \Rightarrow a + d = 6 . . . . . . . (i)\]
\[\text { Now, according to the question, a + 4, a + d + 4 and a + 2d + 36 are in G . P .} \]
\[ \therefore \left( a + d + 4 \right)^2 = \left( a + 4 \right)\left( a + 2d + 36 \right)\]
\[ \Rightarrow \left( 6 - d + d + 4 \right)^2 = \left( 6 - d + 4 \right) \left( 6 - d + 2d + 36 \right) \]
\[ \Rightarrow \left( 10 \right)^2 = \left( 10 - d \right)\left( 42 + d \right)\]
\[ \Rightarrow 100 = 420 + 10d - 42d - d^2 \]
\[ \Rightarrow d^2 + 32d - 320 = 0\]
\[ \Rightarrow \left( d + 40 \right)\left( d - 8 \right) = 0\]
\[ \Rightarrow d = 8, - 40\]
\[\text { Now, putting d = 8, - 40 in equation (i), we get, a = - 2, 46, respectively .} \]
\[\text { For a = - 2 and d = 8, we have }: \]
\[ a = - 2 , b = 6 , c = 14\]
\[\text { And, for a = 46 and d = - 40, we have }: \]
\[ a = 46 , b = 6 , c = - 34\]
संबंधित प्रश्न
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
\[a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .\]
Find:
the ninth term of the G.P. 1, 4, 16, 64, ...
Find :
the 8th term of the G.P. 0.3, 0.06, 0.012, ...
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
Find three numbers in G.P. whose sum is 38 and their product is 1728.
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 terms;
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 terms.
Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
Find the sum of the following series:
0.6 + 0.66 + 0.666 + .... to n terms
The ratio of the sum of the first three terms to that of the first 6 terms of a G.P. is 125 : 152. Find the common ratio.
The 4th and 7th terms of a G.P. are \[\frac{1}{27} \text { and } \frac{1}{729}\] respectively. Find the sum of n terms of the G.P.
Let an be the nth term of the G.P. of positive numbers.
Let \[\sum^{100}_{n = 1} a_{2n} = \alpha \text { and } \sum^{100}_{n = 1} a_{2n - 1} = \beta,\] such that α ≠ β. Prove that the common ratio of the G.P. is α/β.
Find the sum of the following serie to infinity:
\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
Find the sum of the following series to infinity:
`1/3+1/5^2 +1/3^3+1/5^4 + 1/3^5 + 1/56+ ...infty`
Find the rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
If pth, qth, rth and sth terms of an A.P. be in G.P., then prove that p − q, q − r, r − s are in G.P.
Insert 5 geometric means between \[\frac{32}{9}\text{and}\frac{81}{2}\] .
Write the product of n geometric means between two numbers a and b.
The two geometric means between the numbers 1 and 64 are
Check whether the following sequence is G.P. If so, write tn.
`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...
For the G.P. if r = − 3 and t6 = 1701, find a.
Which term of the G.P. 5, 25, 125, 625, … is 510?
Find four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1
Find five numbers in G.P. such that their product is 1024 and fifth term is square of the third term.
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
The number of bacteria in a culture doubles every hour. If there were 50 bacteria originally in the culture, how many bacteria will be there at the end of 5th hour?
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after 10 years.
If Sn, S2n, S3n are the sum of n, 2n, 3n terms of a G.P. respectively, then verify that Sn (S3n – S2n) = (S2n – Sn)2.
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Answer the following:
Find the sum of the first 5 terms of the G.P. whose first term is 1 and common ratio is `2/3`
Answer the following:
Find five numbers in G.P. such that their product is 243 and sum of second and fourth number is 10.
The third term of a G.P. is 4, the product of the first five terms is ______.
