If we have a general case, let's say the geometric sequence #delta_n#, with first term #delta_1# and ratio #q#.
Then, the #k#-th term of the sequence, for an integer #k#, is:
#delta_k = delta_1*q^(k-1)#
Using this knowledge, let's calculate the sum of the first #n# terms:
#delta_1+delta_2+delta_3+...+delta_(n-1)+delta_n = S_n#
#delta_1+delta_1q+delta_1q^2+...+delta_1q^(n-2)+delta_1q^(n-1)=S_n#
Now, multiply both sides by #q#:
#color(red)(delta_1q+delta_1q^2+... +delta_1q^(n-1))+delta_1q^n=S_nq#
Notice how the highlighted part is the sum of the first #n#-th terms, without the first term, #delta_1#.
Hence, we have:
#color(red)(S_n-delta_1)+delta_n=S_nq#
Substract #S_n# on both sides.
#delta_n-delta_1 = S_n(q-1)#
#delta_1q^(n-1)-delta_1=S_n(q-1)#
#delta_1(q^(n-1)-1)=S_n(q-1)#
Finally, we have
#color(red)(S_n = delta_1 (q^(n-1)-1)/(q-1))="first term"*("ratio"^"number of terms"-1)/("ratio"-1)#
In our case, the first term is #3#, the ratio/multiplier is #5# and we wish to add #4# terms.
#S_4 = 3*(5^4-1)/(5-1)=468#
