Question

What is the first term of the geometric sequence presented in the table below? n 4 9 an 243 59,049

Answer 1

Reminder:  the formula for a geom. seq. is

a(n) = a(1)*r^(n-1), where a(1) is the first term, n is the counter and r is the common ratio. 

I first noted that 243 is a power of 3; specifically, 243=3^5, or 243=3(3)^4, or 243=(3^2)(3)^(4-1).  Notice that I'm trying here to rewrite 243=3^5 in the form a(n) = a(1)*r^(n-1):    a(4) = a(1)(3)^(4-1), or a(4) = a(1)(3)^3 = 243.  Then by division we find that a(1) = 243/27 = 9.  Is it possible that a(1)=9?

Let's try out our formula   a(n)=9(3)^(n-1).  Steal n=9 and see whether this formula gives u s 59049:

n(9) = 59049 = 9(3)^(9-1), or  9(3)^8.  True or false?  3^8= 6561, and 9(3)^8 = 59049.

YES!  That's correct.

Therefore, the desired formula is

a(n) = 9(3)^(n-1).  The first term, a(1) is 9(3)^(1-1) = 9(3)^0 = 9*1 = 9.