Question

What is the 12th term of the sequence `4, 12, 36` and what is the sum of all 12 terms?

Answer 1

`a_2/a_1=12/4=3`
`a_3/a_2=36/12=3`
`implies`common ratio`=r=3` and the given sequence is geometric sequence.
Nth term of a geometric sequence is given by
`a_n=a_1r^(n-1)`
Where `a_n` is the nth term, `a` is the first term and `n` is the number of terms.
Here `a_1=4` and `r=3`
Put `n=12`
`implies a_12=4*(3)^(12-1)=4*3^11=4*177147=708588`
`implies` 12th term is `708588`.
Sum of geometric series is given by
`Sum=S=(a_1(1-r^n))/(1-r)`
`implies S=(4(1-3^12))/(1-3)=(4(1-531441))/-2=-2(-531440)=1062880`
`implies Sum=1062880`.