For each of the following, decide if the given series is a geometric series. A.
ID: 2894704 • Letter: F
Question
For each of the following, decide if the given series is a geometric series. A. 2 + 2/2 + 2/3 + 2/4 + middot middot middot: Is this a geometric series? A. Yes B. No If this is a geometric series, enter the first term: and the ratio between successive terms: (Enter na for the first term and ratio if this is not a geometric series.) b. 2 + 4x + 8x^2 + 16x^3 + 32x^4 + middot middot middot: Is this a geometric series? A. Yes B. No If this is a geometric series, enter the first term: and the ratio between successive terms: (Enter na for the first term and ratio if this is not a geometric series.) C. 2x^7 + 3x^8 + 4x^9 + 5x^10 + middot middot middot: Is this a geometric series? A. Yes B. No If this is a geometric series, enter the first term: ______ and the ratio between successive terms: ______Explanation / Answer
to be a geometric series we must have : u_n+1 / un = constant
so here :
(a) 2 + 2/2+ 2/3 + 2/4 + ... : u_n+1 / un
1 / 2 = 1/2 , 2/3 / ( 2/2) = 2/3 , 2/4 / ( 2/3) = 3/4 ...The ratio between terms is consistent , so it is geometric series
(b) 2 + 4x + 8x^2 + 16x^3 + 32x^4+ ... : u_n+1 / un
8x^2 /4x = 2 x , 16x^3 / 8x^2 = 2x .....The ratio between terms is consistent , so it is geometric series
( c) 2x^7 + 3x^8 + 4x^9+ 5x^10 + ... : u_n+1 / un
3x^8 / 2x^7 = ( 3/2) x , 4x^9 / 3 x^8 = ( 4/3) x , 5x^10 / 4 x^9 = ( 5/4) x
The ratio between terms is not consistent , so it is not geometric series
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