The first term of a geometric sequence is 3 and the multiplier, or ratio, is 5. What is the sum of the first 4 terms of the sequence?

1 Answer
Apr 30, 2018

Answer:

#S_4 =468 #

Explanation:

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#