हिंदी

For a G.P. if S5 = 1023 , r = 4, Find a - Mathematics and Statistics

Advertisements
Advertisements

प्रश्न

For a G.P. if S5 = 1023 , r = 4, Find a

योग

उत्तर

Sn = `("a"("r"^"n" - 1))/("r" - 1)`

∴ S5 = `("a"("r"^5 - 1))/("r" - 1)` = 1023, where r = 4

∴ `("a"(4^5 - 1))/(4 - 1)` = 1023

∴ `("a"(1024 - 1))/3` = 1023

∴ a = 3

shaalaa.com
  क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
अध्याय 2: Sequences and Series - Exercise 2.2 [पृष्ठ ३१]

APPEARS IN

बालभारती Mathematics and Statistics 2 (Arts and Science) [English] 11 Standard Maharashtra State Board
अध्याय 2 Sequences and Series
Exercise 2.2 | Q 2. (ii) | पृष्ठ ३१

संबंधित प्रश्न

Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.


Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).


Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.


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.


If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in 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.

 

Find:

the 10th term of the G.P.

\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]

 


Which term of the G.P. :

\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]


If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].


The sum of first three terms of a G.P. is \[\frac{39}{10}\] and their product is 1. Find the common ratio and the terms.

 

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.


Find the sum of the following geometric series:

(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;


The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.


Find the sum :

\[\sum^{10}_{n = 1} \left[ \left( \frac{1}{2} \right)^{n - 1} + \left( \frac{1}{5} \right)^{n + 1} \right] .\]


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 S1, S2, ..., Sn are the sums of n terms of n G.P.'s whose first term is 1 in each and common ratios are 1, 2, 3, ..., n respectively, then prove that S1 + S2 + 2S3 + 3S4 + ... (n − 1) Sn = 1n + 2n + 3n + ... + nn.


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 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 G.P., prove that log a, log b, log c are in A.P.


If a, b, c are in G.P., prove that:

(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.


Find the geometric means of the following pairs of number:

2 and 8


If the fifth term of a G.P. is 2, then write the product of its 9 terms.


If pth, qth and rth terms of a G.P. re x, y, z respectively, then write the value of xq − r yr − pzp − q.

 

 

 


The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. is 


If the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms, then its first term is 


If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term 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 the G.P. if r = `1/3`, a = 9 find t7


For the G.P. if a = `2/3`, t6 = 162, find r.


If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.


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"` = P


Express the following recurring decimal as a rational number:

`2.3bar(5)`


If the A.M. of two numbers exceeds their G.M. by 2 and their H.M. by `18/5`, find the numbers.


Answer the following:

If for a G.P. t3 = `1/3`, t6 = `1/81` find r


If a, b, c, d are in G.P., prove that a2 – b2, b2 – c2, c2 – d2 are also in G.P.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×