find sum of 1 - 1/7 + 1/49 - ... / express 0.7171... as frac

[adambinch]

Can anyone please help me with following:

how can I find the sum of the first n terms of the series

i) 1 - 1/7 + 1/49 - 1/343 + ........

ii) how do I express the recurring decimal 0.717171..... as a fraction

iii) how do I find the sum of the first 51 terms of an arithmetic series which has 45 as its 5th term and 173 as its 21st term

 

[stapel]

Can anyone please help me with following:

how can I find the sum of the first n terms of the series

i) 1 - 1/7 + 1/49 - 1/343 + ........

Find the common ratio, and then apply the finite-sum formula they gave you.

ii) how do I express the recurring decimal 0.717171..... as a fraction

There are various methods. In this context, try decomposing the decimal into fractions corresponding to 0.71, 0.0071, 0.000071, etc. Find the common ratio, and then apply the infinite-sum formula they gave you.

iii) how do I find the sum of the first 51 terms of an arithmetic series which has 45 as its 5th term and 173 as its 21st term

Plug the values of the fifth and twenty-first terms into the formula for the value of the n-th term of an arithmetic series. Find the common difference and initial value. Then plug these into the sum formula they gave you.

If you get stuck, please reply showing all of your work and reasoning so far, including the formulas you are using. Thank you!

Eliz.

 

[YourTutorOnline]

to go from 0.7171 to a fraction:

let x = 0.7171...
100x = 71.717... (multiply both sides by 10[sup:53lwpd02]x[/sup:53lwpd02] where x is the number of times the decimal repeats)
-x - 0.717 (subtract x from both sides)
---- ----------
99x = 71
x = 71/99 (divide both sides by 99. You can verify this answer by typing it into a calculator)

 

[Dr. Flim-Flam]

Both are geometric progressions, and r <1 so Sum = a/(1-r), a being the first term and r being the ratio.

Hence sum of 1 -1/7 + 1/7^2 - 1/7^3 + 1/7^4 - 1/7^5 + ... + 1/7^n = 1/[(1-(-1/7)] = 1/(8/7) = 7/8 = .875.

.71717171... = .71 + .0071 + .000071 + .00000071 + ... Ergo Sum = a/(1-r) = .71/(1-.01) = .71/.99 = 71/99.