Advertisements
Advertisements
Question
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
Advertisements
Solution
Let the required numbers be G1 and G2.
∴ 1, G1, G2, −27 are in G.P.
∴ t1 = 1, t2 = G1, t3 = G2, t4 = −27
∴ t1 = a = 1
tn = arn−1
∴ t4 = (1)r4−1
∴ −27 = r3
∴ r3 = (− 3)3
∴ r = − 3
∴ G1 = t2 = ar = 1(−3) = −3
G2 = t3 = ar2 = 1(−3)2 = 9
∴ For resulting sequence to be G.P. we need to insert numbers −3 and 9.
APPEARS IN
RELATED QUESTIONS
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
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 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, ...
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 terms;
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 terms.
Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n terms.
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?
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`
Express the recurring decimal 0.125125125 ... as a rational number.
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.
Three numbers are in A.P. and their sum is 15. If 1, 3, 9 be added to them respectively, they form a G.P. Find the numbers.
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:
(b + c) (b + d) = (c + a) (c + d)
If a, b, c are in G.P., prove that the following is also in G.P.:
a3, b3, c3
If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.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 a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x< −1 or x > 3.
Find the geometric means of the following pairs of number:
2 and 8
If A1, A2 be two AM's and G1, G2 be two GM's between a and b, then find the value of \[\frac{A_1 + A_2}{G_1 G_2}\]
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 a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then
Check whether the following sequence is G.P. If so, write tn.
`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...
Find four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1
For a G.P. if a = 2, r = 3, Sn = 242 find n
For a G.P. If t3 = 20 , t6 = 160 , find S7
If one invests Rs. 10,000 in a bank at a rate of interest 8% per annum, how long does it take to double the money by compound interest? [(1.08)5 = 1.47]
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
Select the correct answer from the given alternative.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
Answer the following:
Find k so that k – 1, k, k + 2 are consecutive terms of a 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 ______.
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
Let `{a_n}_(n = 0)^∞` be a sequence such that a0 = a1 = 0 and an+2 = 2an+1 – an + 1 for all n ≥ 0. Then, `sum_(n = 2)^∞ a^n/7^n` is equal to ______.
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 ______.
