Question

0.6+0.66+0.666+ ...... 10 terms .

Find the sum of 10 terms .​

Answer 1

Answer:

$0.6+0.66+0.666+........ 10\:times\\\\\implies 6[0.1+0.11+....10\:times]\\\\\implies \frac{6}{9}[0.9+0.99+...10\:times]\\\\\implies \frac{6}{9}[1-0.1+1-0.01+...10\:times]\\\\\implies \frac{6}{9}[(1+1+1..10\:times) -(0.1+0.01+...10\:times)]\\\\\implies \frac{6}{9}[10-\frac{0.1(1-(0.1)^{10}}{1-0.1}]\\\\\implies \frac{2}{3}[10-\frac{1}{10}(\frac{1-\frac{1}{10^{10}}}{\frac{9}{10}})]$

$\implies \frac{2}{3}[10-\frac{1}{10}(\frac{10^{10}-1}{9\times 10^{9}})]\\\\\implies \boxed{\frac{2}{3}[10-(\frac{10^{10}-1}{9\times 10^{10}})]}$

Step-by-step explanation:

The given expression is not in G.P neither in A.P .

So we need to make this expression a G.P .

Start by taking commons .

Take the common and divide every term by 9 .

Then subtract from the nearest tens digit to convert the data into G.P .

This is the method used.

After the conversion to G.P we are not done yet .

We need to group the data and arrange it .

In the given problem we have added 10 times 1 which is equivalent to writing  10 × 1 = 10 .

There after we have solved using the G.P formula :

Sum of n terms of a G.P is given by the formula :

$S_n=a\frac{1-r^n}{1-r}$

Here a is the first term .

n is the number of terms .

r is the common ratio .

Answer 2

ANSWER

Sn = 0.6 + 0.66 + 0.666 +..... Till 10 terms

=>Sn=6(0.1 + 0.11 + 0.111 +..... 10terms)

Now multiplying and dividing by 9 we get>>

Sn=6/9(0.9 + 0.99 + 0.999 +.......10terms)

=>Sn=2/3(0.9+ 0.99 + 0.999 +.......10terms)

=>Sn = 2/3{(1-0.1) + (1-0.11) + (1-0.111) +...... 10 terms}

=>Sn = 2/3[(1+1+1..) - (0.1 + 0.11 + 0.111+.....]

- - - - - - - - - - - (i)

Now, we have

1+1+1+..... N terms = Nn

- - - - - - - (ii)

And

0.1 + 0.11 + 0.111..... N terms

=>0.1{1-(0.1)^n}/(1-0.1)

=>(10^n - 1)/9-10^n

- - - - - - - - - - (iii)

So, from (i), (ii) and(iii) we get >>

Sn = (2n/3) - [2(10^n-1)/27 ×10^n]

Where n = 10.

Some explanation and more information

1)These terms are not in A. P or G.P, we actually divided and multiplied them by 9 to make them into an G. P

2)Geometric progression (G.P) or geometric sequence is an mathematical sequence where next term is found by multiplying the previous one by an fixed, non zero number which is also called a common ratio.
Example =>

3,9,27,81.......

Here each next term is found by multiplying the preceeding term by 3.



3) Arithmetic profession(AP) or an arithmetic sequence is an mathematical sequence where each term is found by adding a particular or constant term to it.


Example >>

6,8,10,12,14....

Where each term is found by adding 2 to the earlier or preceeding term.