Question 1204431: Determine whether 280 is a term in the G.P 1,3,9,27…
Found 3 solutions by josgarithmetic, math_tutor2020, mananth:
Answer by josgarithmetic(39617)   (Show Source):
You can put this solution on YOUR website! Obviously not. Starting with 3, and then only powers of 3, and you know by LOOKING at 280, this is not a power of 3.
In case you did not know or forgot, beginning with 3, any and all powers of 3 will have sum of the digits being equal to a multiple of 3.
Look at 280.
find sum of 2+8+0.
That sum is 10.
10 is very obviously not a multiple of 3.
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!
Method 1
first term = 1
common ratio = 3 (since 3/1 = 3, 9/3 = 3, and 27/9 = 3, etc)
Triple each term to get the next term. Extend the sequence until reaching 280 or going past it.
1, 3, 9, 27, 81, 243, 729, ...
We do not reach 280.
Tip: Use spreadsheet software to generate that list quickly.
The sequence command in GeoGebra is another route using technology. There are many online tools that can be used as well.
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Method 2
The nth term of a geometric progression (GP) is: a*(r)^(n-1)
a = first term
r = common ratio
Because a = 1 and r = 3, we get 1*(3)^(n-1) or simply 3^(n-1)
Set this equal to 280 and solve for n.
Because the exponent is in the trees, we'll need to log it down.
3^(n-1) = 280
log( 3^(n-1) ) = log(280)
(n-1)*log(3) = log(280)
n-1 = log(280)/log(3)
n-1 = 5.1290065
n = 5.1290065+1
n = 6.1290065
We do not achieve a whole number solution for n, so it is impossible to reach 280 when plugging in whole numbers to 3^(n-1)
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Method 3
Let's assume 280 is a term somewhere in 1,3,9,27,...
If so, then repeatedly dividing 280 over powers of 3 should get us to the terms your teacher gave you.
280/3 = 93.3333
We run into a problem almost immediately.
The non-whole number result would mean that repeated "divide by 3" operations would further push the result into non-whole number territory.
Sure enough,
93.3333/3 = 31.1111
31.1111/3 = 10.3703667
10.3703667/3 = 3.4567889
3.4567889/3 = 1.152262967
The results are somewhat close to 1,3,9,27,... but not an exact match.
This proves that 280 is not part of the geometric sequence 1,3,9,27,...
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Method 4
Perhaps the easiest method of the list. No calculators are needed. No need for helpful tech to extend a list of values.
Notice that the subsequence 3,9,27,... are multiples of 3. So we can use the divisibility by 3 rule.
Add the digits of 280
2+8+0 = 10
Repeat the process of adding the digits
1+0 = 1
The digit sum of 1 is not a multiple of 3, so we know that 280 is not a multiple of 3.
Therefore, it is impossible to reach 280 when writing powers of 3.
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Conclusion: 280 is NOT a term of the GP 1,3,9,27,...
Answer by mananth(16946)  (Show Source):
You can put this solution on YOUR website!
Determine whether 280 is a term in the G.P 1,3,9,27…
a= 1
r=3
An =280
tn= a*r^(n-1)
280 = 1*3^(n-1)
When we calculate, n is not a whole number so it is not a term in the given GP
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