# In a geometric sequence #t_3 = 45# #t_6=1215#, how do you find the first term?

Hence if we work from term 3 to term 6 we may write that

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To find the first term (often denoted as ( a )) of a geometric sequence when given the third term (( t_3 )) and the sixth term (( t_6 )), you can use the formula:

[ t_n = a \cdot r^{(n-1)} ]

where:

- ( t_n ) is the ( n )th term of the sequence,
- ( a ) is the first term of the sequence,
- ( r ) is the common ratio of the sequence, and
- ( n ) is the term number.

Given that ( t_3 = 45 ) and ( t_6 = 1215 ), you have two equations:

[ 45 = a \cdot r^{(3-1)} ] [ 1215 = a \cdot r^{(6-1)} ]

You can divide the second equation by the first to eliminate ( a ):

[ \frac{1215}{45} = \frac{a \cdot r^{5}}{a \cdot r^{2}} ]

[ \frac{27}{1} = r^3 ]

[ r = 3 ]

Now, substitute ( r = 3 ) into one of the equations to find ( a ):

[ 45 = a \cdot 3^{2} ]

[ 45 = a \cdot 9 ]

[ a = \frac{45}{9} ]

[ a = 5 ]

So, the first term of the geometric sequence is ( a = 5 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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