Advertisements
Advertisements
Question
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the perimeters of all the squares
Advertisements
Solution
Perimeter of 1st square = 4
Perimeter of 2nd square = `4(1/sqrt(2))`
Perimeter of 3rd square = `4(1/2)`
and so on.
∴ Sum of the perimeters of all the squares
= `4 + 4(1/sqrt(2)) + 4(1/2) + ...`
= `4(1 + 1/sqrt(2) + (1/sqrt(2))^2 + ...)`
The terms `1,1/sqrt(2), (1/sqrt(2))^2, ...` are in G.P.
∴ a = 1, r = `1/sqrt(2)`
Since, |r| = `|1/sqrt(2)| < 1`
∴ sum to infinity exists.
∴ Sum of the perimeters of all the squares
= `4(1/(1 - 1/sqrt(2)))`
= `(4sqrt(2))/(sqrt(2) - 1)`
APPEARS IN
RELATED QUESTIONS
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
If the first and the nth term of a G.P. are a ad b, respectively, and if P is the product of n terms, prove that P2 = (ab)n.
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.
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.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.
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;
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.
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.
Let an be the nth term of the G.P. of positive numbers.
Let \[\sum^{100}_{n = 1} a_{2n} = \alpha \text { and } \sum^{100}_{n = 1} a_{2n - 1} = \beta,\] such that α ≠ β. Prove that the common ratio of the G.P. is α/β.
Find the sum of the following serie to infinity:
`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`
Prove that: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
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.
If a, b, c, d are in G.P., prove that:
\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]
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), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
Insert 5 geometric means between \[\frac{32}{9}\text{and}\frac{81}{2}\] .
Find the geometric means of the following pairs of number:
−8 and −2
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
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 p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
The two geometric means between the numbers 1 and 64 are
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
Find four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1
Find five numbers in G.P. such that their product is 1024 and fifth term is square of the third term.
The number of bacteria in a culture doubles every hour. If there were 50 bacteria originally in the culture, how many bacteria will be there at the end of 5th hour?
Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after n years.
For a G.P. If t3 = 20 , t6 = 160 , find S7
Express the following recurring decimal as a rational number:
`0.bar(7)`
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.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
Select the correct answer from the given alternative.
Which of the following is not true, where A, G, H are the AM, GM, HM of a and b respectively. (a, b > 0)
If pth, qth, and rth terms of an A.P. and G.P. are both a, b and c respectively, show that ab–c . bc – a . ca – b = 1
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.
For an increasing G.P. a1, a2 , a3 ........., an, if a6 = 4a4, a9 – a7 = 192, then the value of `sum_(i = 1)^∞ 1/a_i` is ______.
The sum of infinite number of terms of a decreasing G.P. is 4 and the sum of the terms to m squares of its terms to infinity is `16/3`, then the G.P. is ______.
