SOLUTION: Does the infinite series diverge or converge? If it converges what is the sum? 10+2+2/5+2/25+.... what is a_n= 2a_n-1 -1 where a_1 = 2? Thank you !!!

Algebra ->  Sequences-and-series -> SOLUTION: Does the infinite series diverge or converge? If it converges what is the sum? 10+2+2/5+2/25+.... what is a_n= 2a_n-1 -1 where a_1 = 2? Thank you !!!       Log On


   

Question 1001038: Does the infinite series diverge or converge? If it converges what is the sum?
10+2+2/5+2/25+....
what is
a_n= 2a_n-1 -1 where a_1 = 2?
Thank you !!!
Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
.
Does the infinite series diverge or converge? If it converges what is the sum?
10+2+2/5+2/25+....
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It is the sum a geometric progression with the first term a = 10 and the common ratio r = 1%2F5.

This infinite series converges for any geometric progression with the common ratio less than 1 in the modulus.

The sum of an infinite geometric progression with the first term a and the raio r, |r| < 1, is

S = a%2F%281-r%29.

In your case S = 10%2F%281+-+%281%2F5%29%29 = 10%2F%28%284%2F5%29%29 = %2810%2A5%29%2F4 = 12.5.

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what is
a_n= 2a_n-1 -1 where a_1 = 2?
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a%5B1%5D = 2
a%5B2%5D = 2%2Aa%5B1%5D+-+1 = 2*2 - 1 = 3,
a%5B3%5D = 2%2Aa%5B2%5D+-+1 = 2*3 - 1 = 5,
a%5B4%5D = 2%2Aa%5B3%5D+-+1 = 2*5 - 1 = 9,
a%5B5%5D = 2%2Aa%5B4%5D+-+1 = 2*9 - 1 = 17,
and so on . . .

You can easily calculate it yourself.