Advertisements
Advertisements
प्रश्न
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.
उत्तर
\[a_4 = x\]
\[ \Rightarrow a r^3 = x\]
\[\text { Also }, a_{10} = y\]
\[ \Rightarrow a r^9 = y\]
\[\text { And, } a_{16} = z\]
\[ \Rightarrow a r^{15} = z\]
\[ \because \frac{y}{x} = \frac{a r^9}{a r^3} = r^6 \]
\[\text { and } \frac{z}{y} = \frac{a r^{15}}{a r^9} = r^6 \]
\[ \therefore \frac{y}{x} = \frac{z}{y}\]
\[\text { Therefore, x, y and z are in G . P } .\]
APPEARS IN
संबंधित प्रश्न
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
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 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
Find the sum of the products of the corresponding terms of the sequences `2, 4, 8, 16, 32 and 128, 32, 8, 2, 1/2`
If the first and the nth term of a G.P. are a ad b, respectively, and if P is the product of n terms, prove that P2 = (ab)n.
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.
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.
If a, b, c are in A.P,; b, c, d are in G.P and ` 1/c, 1/d,1/e` are in A.P. prove that a, c, e are in G.P.
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 progression:
4, 2, 1, 1/2 ... to 10 terms.
Find the sum of the following geometric series:
\[\sqrt{7}, \sqrt{21}, 3\sqrt{7}, . . .\text { to n terms }\]
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
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 serie to infinity:
\[\frac{1}{3} + \frac{1}{5^2} + \frac{1}{3^3} + \frac{1}{5^4} + \frac{1}{3^5} + \frac{1}{56} + . . . \infty\]
Insert 5 geometric means between \[\frac{32}{9}\text{and}\frac{81}{2}\] .
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
Write the product of n geometric means between two numbers a and b.
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
If the first term of a G.P. a1, a2, a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of G.P. is
The product (32), (32)1/6 (32)1/36 ... to ∞ is equal to
For the G.P. if r = − 3 and t6 = 1701, find a.
Which term of the G.P. 5, 25, 125, 625, … is 510?
For the following G.P.s, find Sn.
p, q, `"q"^2/"p", "q"^3/"p"^2,` ...
If S, P, R are the sum, product, and sum of the reciprocals of n terms of a G.P. respectively, then verify that `["S"/"R"]^"n"` = P2
Find: `sum_("r" = 1)^10 5 xx 3^"r"`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`-3, 1, (-1)/3, 1/9, ...`
Express the following recurring decimal as a rational number:
`2.bar(4)`
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
Find : `sum_("n" = 1)^oo 0.4^"n"`
Answer the following:
Find `sum_("r" = 1)^"n" (2/3)^"r"`
Answer the following:
If for a G.P. first term is (27)2 and seventh term is (8)2, find S8
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
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.
Answer the following:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
The third term of a G.P. is 4, the product of the first five terms is ______.
