Advertisements
Advertisements
प्रश्न
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
Advertisements
उत्तर
\[\text { Let the 6 G . M . s between 27 and } \frac{1}{81}\text { be} G_1 , G_2 , G_3 , G_4 , G_5 \text { and } G_6 . \]
\[\text { Thus }, 27, G_1 , G_2 , G_3 , G_4 , G_5 , G_6 \text { and } \frac{1}{81} \text { are in G . P } . \]
\[ \therefore a = 27, n = 8 \text { and } a_8 = \frac{1}{81}\]
\[ \because a_8 = \frac{1}{81}\]
\[ \Rightarrow {ar}^7 = \frac{1}{81}\]
\[ \Rightarrow r^7 = \frac{1}{81 \times 27}\]
\[ \Rightarrow r^7 = \left( \frac{1}{3} \right)^7 \]
\[ \Rightarrow r = \frac{1}{3}\]
\[ \therefore G_1 = a_2 = ar = 27\left( \frac{1}{3} \right) = 9\]
\[ G_2 = a_3 = a r^2 = 27 \left( \frac{1}{3} \right)^2 = 3\]
\[ G_3 = a_4 = a r^3 = 27 \left( \frac{1}{3} \right)^3 = 1\]
\[ G_4 = a_5 = a r^4 = 27 \left( \frac{1}{3} \right)^4 = \frac{1}{3}\]
\[ G_5 = a_6 = a r^5 = 27 \left( \frac{1}{3} \right)^5 = \frac{1}{9} \]
\[ G_6 = a_7 = a r^6 = 27 \left( \frac{1}{3} \right)^6 = \frac{1}{27}\]
APPEARS IN
संबंधित प्रश्न
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
Find the sum to indicated number of terms in the geometric progressions x3, x5, x7, ... n terms (if x ≠ ± 1).
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`.
if `(a+ bx)/(a - bx) = (b +cx)/(b - cx) = (c + dx)/(c- dx) (x != 0)` then show that a, b, c and d are in G.P.
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
Which term of the G.P. :
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.
Find the sum of the following geometric series:
`sqrt7, sqrt21, 3sqrt7,...` to n terms
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
Find the sum of the following serie:
5 + 55 + 555 + ... 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.
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 rational numbers having the following decimal expansion:
\[0 . 6\overline8\]
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.
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
If a, b, c, d are in G.P., prove that:
\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]
If a, b, c are in G.P., then prove that:
If \[\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}\] are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
Check whether the following sequence is G.P. If so, write tn.
2, 6, 18, 54, …
For the G.P. if a = `7/243`, r = 3 find t6.
Find five numbers in G.P. such that their product is 1024 and fifth term is square of the third term.
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.
Find: `sum_("r" = 1)^10 5 xx 3^"r"`
Express the following recurring decimal as a rational number:
`2.3bar(5)`
Express the following recurring decimal as a rational number:
`51.0bar(2)`
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the areas of all the squares
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
Answer the following:
If for a G.P. t3 = `1/3`, t6 = `1/81` find r
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.
Answer the following:
If p, q, r, s are in G.P., show that (pn + qn), (qn + rn) , (rn + sn) are also in G.P.
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series `sqrt(x)(sqrt(a) + sqrt(x)) + sqrt(x)(sqrt(ab) + sqrt(xy)) + sqrt(x)(bsqrt(a) + ysqrt(x)) + ...` is ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
