Advertisements
Advertisements
प्रश्न
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.
उत्तर
Let a be the first term and r be the common ratio of the G.P.
\[a_2 = 24 \]
\[ \Rightarrow a r^{2 - 1} = 24\]
\[ \Rightarrow ar = 24 . . . \left( i \right)\]
\[\text { Similarly }, a_5 = 81 \]
\[ \Rightarrow a r^{5 - 1} = 24\]
\[ \Rightarrow a r^4 = 81\]
\[ \Rightarrow \frac{24 \times r^4}{r} = 81 \left[ \text { From } \left( i \right) \right]\]
\[ \Rightarrow r^3 = \frac{81}{24} \]
\[ \therefore r^3 = \frac{27}{8}\]
\[ \Rightarrow r = \frac{3}{2}\]
\[\text { Putting }r = \frac{3}{2}\text { in } \left( i \right)\]
\[3a = 48 \]
\[ \Rightarrow a = 16\]
\[\text { So, the geometric series is } 16 + 24 + 36 + . . . + 16 \left( \frac{3}{2} \right)^8 \]
\[\text { And }, S_8 = 16\left( \frac{\left( \frac{3}{2} \right)^8 - 1}{\frac{3}{2} - 1} \right) \]
\[ \Rightarrow S_8 = 32\left( \frac{6561 - 256}{256} \right) = \frac{32 \times 6305}{256} = \frac{6305}{8}\]
APPEARS IN
संबंधित प्रश्न
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
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 `1/r^n`.
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.
Which term of the G.P. :
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]
The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is \[87\frac{1}{2}\] . Find them.
The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.
Evaluate the following:
\[\sum^{11}_{n = 1} (2 + 3^n )\]
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n terms.
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 \[\frac{1}{r^n}\].
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 sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
If S denotes the sum of an infinite G.P. S1 denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively
\[\frac{2S S_1}{S^2 + S_1}\text { and } \frac{S^2 - S_1}{S^2 + S_1}\]
If a, b, c are in A.P. and a, b, d are in G.P., then prove that a, a − b, d − c are in G.P.
If pth, qth, rth and sth terms of an A.P. be in G.P., then prove that p − q, q − r, r − s are in G.P.
If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
Find the geometric means of the following pairs of number:
−8 and −2
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
If x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
Find four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1
The numbers 3, x, and x + 6 form are in G.P. Find x
For a G.P. if S5 = 1023 , r = 4, Find a
The value of a house appreciates 5% per year. How much is the house worth after 6 years if its current worth is ₹ 15 Lac. [Given: (1.05)5 = 1.28, (1.05)6 = 1.34]
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
If the first term of the G.P. is 6 and its sum to infinity is `96/17` find the common ratio.
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
Find : `sum_("r" = 1)^oo (-1/3)^"r"`
Select the correct answer from the given alternative.
The common ratio for the G.P. 0.12, 0.24, 0.48, is –
Select the correct answer from the given alternative.
If for a G.P. `"t"_6/"t"_3 = 1458/54` then r = ?
Select the correct answer from the given alternative.
If common ratio of the G.P is 5, 5th term is 1875, the first term is -
Answer the following:
Find the sum of the first 5 terms of the G.P. whose first term is 1 and common ratio is `2/3`
Answer the following:
For a sequence , if tn = `(5^("n" - 2))/(7^("n" - 3))`, verify whether the sequence is a G.P. If it is a G.P., find its first term and the common ratio.
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
At the end of each year the value of a certain machine has depreciated by 20% of its value at the beginning of that year. If its initial value was Rs 1250, find the value at the end of 5 years.
The sum or difference of two G.P.s, is again a G.P.
