Advertisements
Advertisements
प्रश्न
Prove that: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
Advertisements
उत्तर
\[\text { LHS } = 2^\frac{1}{4} . 4^\frac{2}{8} . 8^\frac{3}{16} . {16}^\frac{4}{32} . . . \infty \]
\[ = 2^\left( \frac{1}{4} + \frac{2}{8} + \frac{3}{16}\frac{3}{16}\frac{4}{32} . \infty \right) \]
\[ = 2^\left( \frac{1}{2^2} + \frac{2}{2^3} + \frac{3}{2^4} + \frac{4}{2^5} + . . . \infty \right) \]
\[ = 2^\frac{1}{2^2}\left\{ 1 + \frac{2}{2} + \frac{3}{2^2} + \frac{4}{2^3} . . . \infty \right\} \]
\[ = 2^\frac{1}{2^2}\left\{ \frac{1}{1 - \frac{1}{2}} + \frac{1 . \frac{1}{2}}{\left( 1 - \frac{1}{2} \right)^2} \right\} \]
\[ = 2^\frac{1}{2^2}\left\{ 2 + 2 \right\} \]
\[ = 2^1 \]
\[ = 2 = \text { RHS }\]
APPEARS IN
संबंधित प्रश्न
The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that `a^(q - r) b^(r-p) c^(p-q) = 1`.
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
Find:
the ninth term of the G.P. 1, 4, 16, 64, ...
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
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.
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n terms.
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
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.
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:
`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`
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`
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 xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
Insert 5 geometric means between \[\frac{32}{9}\text{and}\frac{81}{2}\] .
Find the geometric means of the following pairs of number:
−8 and −2
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
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
The nth term of a G.P. is 128 and the sum of its n terms is 255. If its common ratio is 2, then its first term is ______.
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 = `1/3`, a = 9 find t7
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
Answer the following:
For a sequence , if tn = `(5^("n" - 2))/(7^("n" - 3))`, verify whether the sequence is a G.P. If it is a G.P., find its first term and the common ratio.
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
Answer the following:
Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
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
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
In a G.P. of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the G.P. is ______.
The third term of G.P. is 4. The product of its first 5 terms is ______.
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
