Advertisements
Advertisements
प्रश्न
Answer the following:
If pth, qth and rth terms of a G.P. are x, y, z respectively. Find the value of xq–r .yr–p .zp–q
Advertisements
उत्तर
Let A be the first term and R be the common ratio of the G.P.
Then tn = ARn–1
Now, tp = x, tq = y and tr = z
∴ ARp–1 = x, ARq–1 = y and ARr–1 = z
∴ xq–r .yr–p .zp–q
= (ARp–1)q–r . (ARq–1)r–p . (ARr–1)p–q
`="A"^("q"–"r") * "R"^("pq"–"pr"–"q"+"r") * "A"^("r"–"p") * "R"^("qr"–"pq"-"r"+"p") * "A"^("p"–"q") *"R"^("pr"–"qr"–"p"+"q")`
= `"A"^("q"–"r"+"r"–"p"+"p"-"q") * "R"^("pq"–"pr"–"q"+"r"+"qr"–"pq"–"r"+"p"+"pr"–"qr"–""p"+"q")`
= A° · R°
= 1
APPEARS IN
संबंधित प्रश्न
Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
Show that one of the following progression is a G.P. Also, find the common ratio in case:
\[a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .\]
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is \[87\frac{1}{2}\] . Find them.
The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.
Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
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.
The fifth term of a G.P. is 81 whereas its second term is 24. Find the series and sum of its first eight terms.
If a and b are the roots of x2 − 3x + p = 0 and c, d are the roots x2 − 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q − p) = 17 : 15.
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
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 an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
The sum of three numbers which are consecutive terms of an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.
If a, b, c are in G.P., prove that:
\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]
If a, b, c, d are in G.P., prove that:
\[\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .\]
If the 4th, 10th and 16th terms of a G.P. are x, y and z respectively. Prove that x, y, z are in G.P.
Find the geometric means of the following pairs of number:
−8 and −2
The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio `(3+2sqrt2):(3-2sqrt2)`.
If the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms, then its first term is
If a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then
If x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
For the G.P. if r = − 3 and t6 = 1701, find a.
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
The numbers x − 6, 2x and x2 are in G.P. Find nth term
For the following G.P.s, find Sn.
p, q, `"q"^2/"p", "q"^3/"p"^2,` ...
For a G.P. If t3 = 20 , t6 = 160 , find S7
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`-3, 1, (-1)/3, 1/9, ...`
Find : `sum_("n" = 1)^oo 0.4^"n"`
Select the correct answer from the given alternative.
Sum to infinity of a G.P. 5, `-5/2, 5/4, -5/8, 5/16,...` 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 the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
Answer the following:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
