Back to Tutor Intake Assessment: Derive geometric series formula

HSA.SSE.B.4 Tutor Intake — Geometric Series Formula

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Grade 9·10 problems·~15 min·Common Core Math - HS Algebra·standard·hsa-sse-b-4

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Concepts

Consider the geometric series $2 + 6 + 18 + 54$ .

What does the formula $S = \dfrac{a(1 - r^n)}{1 - r}$ compute for
this series?

The 4th term of the sequence ( $ar^3 = 54$ )

The sum of all 4 terms ( $2 + 6 + 18 + 54 = 80$ )

The common ratio ( $r = 3$ )

The number of terms ( $n = 4$ )

For the geometric series $5 + 15 + 45 + 135$ , what are the values
of $a$ , $r$ , and $n$ ?

$a = 5$ , $r = 3$ , $n = 4$

$a = 5$ , $r = 3$ , $n = 135$

$a = 5$ , $r = 3$ , $n = 3$

$a = 15$ , $r = 3$ , $n = 4$

A student wants to find the sum of $7 + 7 + 7 + 7 + 7$ (five 7s).
They try to apply the formula $S = \dfrac{a(1-r^n)}{1-r}$ with
$a = 7$ , $r = 1$ , $n = 5$ .

What should the student do instead, and why?

Use $S = na = 5 \times 7 = 35$ , because the formula is undefined when $r = 1$ (divides by zero)

Substitute $r = 1$ into the formula to get $S = 7(1-1)/0 = 0$

Use $S = ar^n = 7 \times 1^5 = 7$

The formula works fine; the answer is $5 \times 7 = 35$ using the formula

Procedures

For the geometric series $1 + 3 + 9 + 27 + 81 + 243$ ,
identify the number of terms $n$ .

Enter the value of $n$ .

Find the sum of the geometric series $1 + 3 + 9 + 27 + 81 + 243$
using the formula $S = \dfrac{a(r^n - 1)}{r - 1}$ with
$a = 1$ , $r = 3$ , $n = 6$ .

Enter the sum $S$ .

Find the sum of the geometric series
$64 + 32 + 16 + 8 + 4 + 2 + 1$
using the formula $S = \dfrac{a(1 - r^n)}{1 - r}$ with
$a = 64$ , $r = \dfrac{1}{2}$ , $n = 7$ .

Enter the sum $S$ .

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