If the first term of a geometric sequence is $u_1$u1 and the common ratio is $r$r, then the sequence is given by:
$u_1,u_1r,u_1r^2,u_1r^3,u_1r^4,\ldots$u1,u1r,u1r2,u1r3,u1r4,…
Suppose we wish to add the first $n$n terms of this sequence. This will form a geometric series. We could write the sum as:
$S_n=u_1+u_1r+u_1r^2+u_1r^3+...+u_1r^{n-1}$Sn=u1+u1r+u1r2+u1r3+...+u1rn−1
We saw a nifty trick for finding the formula for an arithmetic series by adding two sums together and pairing up terms. We will use a similar method here. If we multiply both sides of our geometric sum by the common ratio $r$r we see that:
$rS_n=u_1r+u_1r^2+u_1r^3+...+u_1r^{n-1}+u_1r^n$rSn=u1r+u1r2+u1r3+...+u1rn−1+u1rn
Then, by carefully subtracting $rS_n$rSn from $S_n$Sn term by term, we see that all of the middle terms disappear:
$S_n-rS_n=u_1+\left(u_1r-u_1r\right)+\left(u_1r^2-u_1r^2\right)+...+\left(u_1r^{n-1}-u_1r^{n-1}\right)-u_1r^n$Sn−rSn=u1+(u1r−u1r)+(u1r2−u1r2)+...+(u1rn−1−u1rn−1)−u1rn
This means that:
$S_n-rS_n=u_1-u_1r^n$Sn−rSn=u1−u1rn
and when common factors are taken out on both sides of this equation, we find:
$S_n\left(1-r\right)=u_1\left(1-r^n\right)$Sn(1−r)=u1(1−rn)
Finally, by dividing both sides by $\left(1-r\right)$(1−r) (excluding the trivial case of $r=1$r=1) we reveal the geometric sum formula:
$S_n=\frac{u_1\left(1-r^n\right)}{1-r}$Sn=u1(1−rn)1−r
An extra step, multiplying the numerator and denominator by $-1$−1, reveals a different form for $S_n$Sn. Both formulas will work in any situation, particularly when using a calculator. This form is generally easier to manage when the common ratio is greater than $r=1$r=1:
$S_n=\frac{u_1\left(r^n-1\right)}{r-1}$Sn=u1(rn−1)r−1
For any geometric sequence with starting value $u_1$u1 and common ratio $r$r, we can find the sum of the first $n$n terms, using:
$S_n=\frac{u_1\left(1-r^n\right)}{1-r}$Sn=u1(1−rn)1−r, for $r<1$r<1 or $S_n=\frac{u_1\left(r^n-1\right)}{r-1}$Sn=u1(rn−1)r−1, convenient if $r>1$r>1
The formulas for $S_n$Sn exclude the case for $r=1$r=1. For the case where $r=1$r=1, then the sequence becomes $u_1,u_1,u_1,u_1,\ldots$u1,u1,u1,u1,….
Is this a geometric sequence with $r=1$r=1 or an arithmetic sequence with $d=0$d=0? Either way, every term is clearly identical. Hence, the sum of the first $n$n terms is:
$S_n$Sn | $=$= | $u_1+u_1+u_1+...+u_1$u1+u1+u1+...+u1 | ($n$n times) |
$S_n$Sn | $=$= | $nu_1$nu1 |
If the sum for the first $n$n terms of the geometric sequence $5,10,20,\ldots$5,10,20,… is $5115$5115, find $n$n.
Think: We have an increasing geometric sequence. State $u_1$u1, $r$r and $S_n$Sn, then substitute into the formula $S_n=\frac{u_1\left(r^n-1\right)}{r-1}$Sn=u1(rn−1)r−1 and rearrange.
Do: $u_1=5$u1=5, $r=2$r=2 and $S_n=5115$Sn=5115, so we have:
$S_n$Sn | $=$= | $\frac{u_1\left(r^n-1\right)}{r-1}$u1(rn−1)r−1 | |
$5115$5115 | $=$= | $\frac{5\left(2^n-1\right)}{2-1}$5(2n−1)2−1 |
Substitute values into formula
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Hence,$5\left(2^n-1\right)$5(2n−1) | $=$= | $5115$5115 |
Simplify fraction and bring unknown to left-hand side
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$2^n-1$2n−1 | $=$= | $1023$1023 |
Divide both sides by $5$5
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$2^n$2n | $=$= | $1024$1024 |
Add $1$1 to both sides
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$\therefore n$∴n | $=$= | $10$10 |
Solve for $n$n, using guess and check, technology or logarithms.
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Consider the series $2+8+32$2+8+32 ...
Find the sum of the first $8$8 terms.
The sum of the first $8$8 terms of a geometric series is $82$82 times the sum of its first $4$4 terms. Solve for $r$r, the common ratio.
Consider the series $2-10+50-\text{. . . }-3906250$2−10+50−. . . −3906250.
Solve for $n$n, the number of terms in the series.
Find the sum of the series.
Average annual salaries are expected to increase by $4$4 percent each year. If the average annual salary this year is found to be $\$43000$$43000:
Calculate the expected average annual salary in $5$5 years, correct to the nearest cent.
This year, Edward starts at a new job in which he will receive the average annual salary for each year of his employment. Over the coming $5$5 years (including this year) he plans to save half of each year’s annual salary.
What will be his total savings over these $5$5 years? Give your answer correct to the nearest cent.