Advertisements
Advertisements
Question
If a, b, c are in A.P. and a, b, d are in G.P., show that a, (a − b), (d − c) are in G.P.
Solution
\[\text { a, b and c are in A . P } . \]
\[ \therefore 2b = a + c . . . . . . . (i)\]
\[\text { Also, a, b and d are in G . P } . \]
\[ \therefore b^2 = ad . . . . . . . (ii)\]
\[\text { Now }, \left( a - b \right)^2 = a^2 - 2ab + b^2 \]
\[ \Rightarrow \left( a - b \right)^2 = a^2 - a\left( a + c \right) + ad \left[ \text { Using } (i)\text { and } (ii) \right]\]
\[ \Rightarrow \left( a - b \right)^2 = a^2 - a^2 - ac + ad\]
\[ \Rightarrow \left( a - b \right)^2 = ad - ac\]
\[ \Rightarrow \left( a - b \right)^2 = a(d - c)\]
\[\text { Therefore, }a, \left( a - b \right) \text { and } (d - c) \text { are in G . P }. \]
APPEARS IN
RELATED QUESTIONS
Which term of the following sequence:
`1/3, 1/9, 1/27`, ...., is `1/19683`?
Evaluate `sum_(k=1)^11 (2+3^k )`
The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.
Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P2Rn = Sn
If a and b are the roots of are roots of x2 – 3x + p = 0 , and c, d are roots of x2 – 12x + q = 0, where a, b, c, d, form a G.P. Prove that (q + p): (q – p) = 17 : 15.
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
Find :
the 12th term of the G.P.
\[\frac{1}{a^3 x^3}, ax, a^5 x^5 , . . .\]
Which term of the G.P. :
\[\sqrt{3}, 3, 3\sqrt{3}, . . . \text { is } 729 ?\]
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
The sum of first three terms of a G.P. is 13/12 and their product is − 1. Find the G.P.
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
Find the sum of the following geometric series:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
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 a, b, c are in G.P., prove that:
a (b2 + c2) = c (a2 + b2)
If a, b, c, d are in G.P., prove that:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2
If a, b, c, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (c2 − d2) are in G.P.
If x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
For the G.P. if a = `2/3`, t6 = 162, find r.
The numbers x − 6, 2x and x2 are in G.P. Find x
For the following G.P.s, find Sn.
p, q, `"q"^2/"p", "q"^3/"p"^2,` ...
For a G.P. If t4 = 16, t9 = 512, find S10
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
Express the following recurring decimal as a rational number:
`0.bar(7)`
Express the following recurring decimal as a rational number:
`2.3bar(5)`
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the 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 or difference of two G.P.s, is again a G.P.
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 in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
