Advertisements
Advertisements
प्रश्न
If a, b, c are in G.P., prove that the following is also in G.P.:
a2 + b2, ab + bc, b2 + c2
Advertisements
उत्तर
a, b and c are in G.P.
∴ \[b^2 = ac . . . . . . . (1)\]
\[\left( ab + bc \right)^2 = \left( ab \right)^2 + 2a b^2 c + \left( bc \right)^2 \]
\[ \Rightarrow \left( ab + bc \right)^2 = \left( ab \right)^2 + a b^2 c + a b^2 c + \left( bc \right)^2 \]
\[ \Rightarrow \left( ab + bc \right)^2 = a^2 b^2 + ac\left( ac \right) + b^2 \left( b^2 \right) + b^2 c^2 \left[ \text { Using } (1) \right]\]
\[ \Rightarrow \left( ab + bc \right)^2 = a^2 \left( b^2 + c^2 \right) + b^2 \left( b^2 + c^2 \right)\]
\[ \Rightarrow \left( ab + bc \right)^2 = \left( b^2 + c^2 \right)\left( a^2 + b^2 \right)\]
\[\text { Therefore }, \left( a^2 + b^2 \right), \left( b^2 + c^2 \right) \text { and }\left( ab + bc \right) \text { are also in G . P } . \]
APPEARS IN
संबंधित प्रश्न
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
Evaluate `sum_(k=1)^11 (2+3^k )`
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.
Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
Find the sum of the following geometric series:
\[\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };\]
Find the sum of the following geometric series:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n terms.
Find the sum of the following series:
0.6 + 0.66 + 0.666 + .... to n terms
The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, find the first term.
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 S1, S2, S3 be respectively the sums of n, 2n, 3n terms of a G.P., then prove that \[S_1^2 + S_2^2\] = S1 (S2 + S3).
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 rational number whose decimal expansion is `0.4bar23`.
Find the rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
One side of an equilateral triangle is 18 cm. The mid-points of its sides are joined to form another triangle whose mid-points, in turn, are joined to form still another triangle. The process is continued indefinitely. Find the sum of the (i) perimeters of all the triangles. (ii) areas of all triangles.
If a, b, c, d are in G.P., prove that:
\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]
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.
If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x2, b2, y2 are in A.P.
If the fifth term of a G.P. is 2, then write the product of its 9 terms.
If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is \[\frac{9}{2}\], then write its first term and common difference.
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
The two geometric means between the numbers 1 and 64 are
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
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?
For the following G.P.s, find Sn
3, 6, 12, 24, ...
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`2, 4/3, 8/9, 16/27, ...`
Express the following recurring decimal as a rational number:
`2.3bar(5)`
Find : `sum_("n" = 1)^oo 0.4^"n"`
In a G.P. of positive terms, if any term is equal to the sum of the next two terms. Then the common ratio of the G.P. is ______.
The sum of the infinite series `1 + 5/6 + 12/6^2 + 22/6^3 + 35/6^4 + 51/6^5 + 70/6^6 + ....` is equal to ______.
If the sum of an infinite GP a, ar, ar2, ar3, ...... . is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, .... is ______.
