Advertisements
Advertisements
Question
If the fifth term of a G.P. is 2, then write the product of its 9 terms.
Advertisements
Solution
Here, a5 = 2
\[\Rightarrow a r^4 = 2\]
Product of the nine terms, i.e.
\[a, ar, a r^2 , a r^3 , a r^4 , a r^5 , a r^6 , a r^7 \text { and } a r^8\]
\[\left( a \times a r^8 \right)\left( ar \times a r^7 \right)\left( a r^2 \times a r^6 \right)\left( a r^3 \times a r^5 \right)\left( a r^4 \right) = \left( a r^4 \right)^9 \]
\[ \because a r^4 = 2\]
\[\text { Required product } = 2^9 = 512\]
RELATED QUESTIONS
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
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.
Find :
the 8th term of the G.P. 0.3, 0.06, 0.012, ...
Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
Find the sum of the following geometric progression:
1, −1/2, 1/4, −1/8, ... to 9 terms;
Find the sum of the following geometric series:
`3/5 + 4/5^2 + 3/5^3 + 4/5^4 + ....` to 2n terms;
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
The 4th and 7th terms of a G.P. are \[\frac{1}{27} \text { and } \frac{1}{729}\] respectively. Find the sum of n terms of the G.P.
Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1)th to (2n)th term is \[\frac{1}{r^n}\].
A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.
Find the sum of the terms of an infinite decreasing G.P. in which all the terms are positive, the first term is 4, and the difference between the third and fifth term is equal to 32/81.
Show that in an infinite G.P. with common ratio r (|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.
If a, b, c are in G.P., prove that log a, log b, log c are in A.P.
If a, b, c are in G.P., prove that:
\[\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}\]
If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
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 x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
Check whether the following sequence is G.P. If so, write tn.
7, 14, 21, 28, …
Which term of the G.P. 5, 25, 125, 625, … is 510?
For a G.P. If t3 = 20 , t6 = 160 , find S7
For a G.P. If t4 = 16, t9 = 512, find S10
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
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:
`-3, 1, (-1)/3, 1/9, ...`
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
Answer the following:
For a sequence Sn = 4(7n – 1) verify that the sequence is a G.P.
Answer the following:
For a G.P. if t2 = 7, t4 = 1575 find a
Answer the following:
Find k so that k – 1, k, k + 2 are consecutive terms of a G.P.
If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is ______.
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 ______.
