Find the value of
V (13-)
1/9
1/13
13
1/81
This is a required question
11Q: *
Answers
Step-by-step explanation:
.
The given series,
1/3+1/9+1/27+1/81+1/243 ...∞
S (for infinite terms) = a₁/(1-r)
So, for the given series,
a₁ = 1/3
& r = aₙ/aₙ₋₁
=> r=(1/9) ÷(1/3)
=> r=1/3
Putting the values in the formula, we get
S=(1/3)/[1–(1/3)]
=> S=1/2=0.5
Ta-da, that's the required answer!
But this approach seems pretty formulae ridden. Doesn't it?
Let me take you to a more interesting and intuitive approach to solve such problems.
in pre-calculus courses.
So, in order to solve this problem, we can take a more systematic approach.
Let's assume, 1/3 + 1/9 + 1/27 + 1/81 + 1/243…∞ = x
On multiplying each side by 3, we get
1 + 1/3 + 1/9 + 1/27 + 1/81… ∞ = 3x
=> 1 + (1/3 + 1/9 + 1/27 + 1/81 + 1/243… ∞)= 3x
On putting (1/3 + 1/9 + 1/27 + 1/81… ∞) = x, we get
=> 1 + x = 3x
=> 1 = 2x
=> x= 1/2= 0.5
Ta-da, once again have got the answer!
Bonus point
Such problems baffled many great think-tanks and philosophers for centuries and came to be regarded as paradoxical problems. One of the famous ones is Zeno's Dichotomy Paradox. Zeno wondered, how come the sum of infinite numbers be finite. He took a realistic approach to explain his paradox using the concepts of time and distances. You can educate yourself more on this here, What is Zeno's Dichotomy Paradox? - Colm Kelleher
Or
Solution:
The given number sequence is 10, 9, 60, 90, 70, 66
We have to find the next number in the sequence, that is to find the number that comes immediately after 66. To do that, we write the numbers in the sequence in words.
Ten, Nine, Sixty, Ninety, Seventy, Sixty Six
and count the number of letters in each of the numerals. They are:
3, 4, 5, 6, 7, 8
It is easily seen that the number of letters form an A.P. with the number of letters in first number ten = 3 and common difference = 1. Therefore the next number after 66 would be one which will have 9 letters. Some of the numerals that have nine letters are: seventeen, twenty one, twenty two, twenty six, thirty one, thirty two, thirty six, fifty four,….sixty four, ninety two, ninety six. Out of the numbers discovered, we will select the greatest (ninety six) since the given sequence is an increasing one and each of the number in the sequence is the greatest of all the numbers that have a particular number of letters. Thus 66 has 8 letters and other numbers having 8 letters are 61, 62. And the one chosen is 66. Same thing with the other numbers 10,9,60,90,70. Therefore, the number that comes next to 66 in the number sequence is: 96 (Answer)