# The air in a hot-air balloon cools as the balloon rises. If the air is not reheated, the balloon rises more slowly every minute. The hot-air balloon rises 50 m in the first minute. In each succeeding minute the balloon rises 70% as far as it did in the previous minute. How far does the balloon rise in 7 minutes, to the nearest meter?

The balloon will rise about 153 m.

`70 % ` of the rise R of the previous minute is `0.7 R . ` The rise of the next minute is `0.7 ( 0.7 R ) = ( 0.7 )^2 R . ` This way, the sequence of rises is

`R , ` `0.7 R , ` `( 0.7 )^2 R , ` ... , `( 0.7 )^( n - 1 ) R , ` ...

Such a sequence is a geometric sequence, with the first term `R ` and the common ratio `0 . 7 .`

Now observe that the height which the balloon gains in n minutes is not the nth term of this sequence but the sum of all its first n terms. Fortunately, the sum of first `n ` terms of a geometric sequence has a relatively short representation:

`S_n = sum_( k = 1 )^( n ) ( ( 0.7 )^( k - 1 ) R ) = R * ( 1 - ( 0.7 )^n ) / ( 1 - 0.7 ) .`

In `7 ` minutes, it will be `5 0 * ( 1 - ( 0.7 )^7 ) / ( 1 - 0.7 ) ` `approx 1 5 3 ( m ) .`

Additionally, the maximum height gain will be not much greater, `50 * ( 1 ) / ( 1 - 0.7 ) approx 167 ( m ) .`

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