Number Sequence

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May 10th 2011, 05:33 AM
Punch

Number Sequence

Deduce the $rth$ term of the series $5+55+555+5555+....$ can be expressed as $\frac{5}{9}(10^r-1)$ .

Hence find, in terms of $k$ , an expression for the sum of the first $k$ terms of the series $5+55+555+5555+....$

I tried to find the common difference or common ratio to see if it is an arithmetic or geometric progression but it didnt work out....
May 10th 2011, 05:53 AM
emakarov

The sequence 5, 55, 555, ... is neither an arithmetic nor a geometric progression. Rather, the rth element is itself (a multiple of) the sum of a geometric progression 5(10^0 + 10^1 + ... + 10^{r-1}).
May 10th 2011, 07:03 AM
chisigma

Quote:

Originally Posted by Punch

Deduce the $rth$ term of the series $5+55+555+5555+....$ can be expressed as $\frac{5}{9}(10^r-1)$ .

Hence find, in terms of $k$ , an expression for the sum of the first $k$ terms of the series $5+55+555+5555+....$

I tried to find the common difference or common ratio to see if it is an arithmetic or geometric progression but it didnt work out....

You have to remember the formula of the sum of n terms of a geometric series...

http://quicklatex.com/cache3/ql_3e5a...5a45cb0_l3.png (1)

... so that the r-th term of your series is obtained from (1) setting http://quicklatex.com/cache3/ql_3e2a...00e8945_l3.png and http://quicklatex.com/cache3/ql_8bdb...1270162_l3.png...

http://quicklatex.com/cache3/ql_edc2...a505c76_l3.png (2)

Now observing (2) You easily conclude that the series is the sum of two series, one 'geometric' so that You can apply again (1) obtaining...

http://quicklatex.com/cache3/ql_75f8...9e11d8c_l3.png (3)

Kind regards

$\chi$ $\sigma$