ACT Math: Calculate the Sum of a Geometric Series Using the Formula

A geometric series is a sum of terms where each term is a constant multiple (the common ratio r) of the previous term. Example: 2+6+18+54 (r=3). Finite series (sum of n terms): S_n=a(1-r^n)/(1-r), where a is the first term and r is the common ratio. Example: Sum of the first 4 terms of 2+6+18+54: S_4=2(1-3⁴)/(1-3)=2(1-81)/(-2)=2(-80)/(-2)=80. Infinite series (if |r|<1): S=a/(1-r). Example: 1+1/2+1/4+1/8+... (a=1, r=1/2). Sum=1/(1-1/2)=1/(1/2)=2. The key is identifying a and r, then applying the right formula (finite or infinite).

Why it matters: Geometric series appear in word problems about repeated growth, savings, decay, and other scenarios where quantities multiply each time. Understanding the formula lets you solve these problems without calculating each term individually.

Mistake 1: Using the infinite series formula when the series is finite (or vice versa). If the problem asks for the sum of the first n terms, use S_n. If it asks for the sum of the series that goes on forever (and |r|<1), use S=a/(1-r). Check the problem statement. Mistake 2: Identifying r incorrectly. r is the ratio between consecutive terms. In 2, 6, 18, 54, the ratio is 6/2=3 (or 18/6=3). If you calculate r as the ratio of the first term to the second, you get 1/3, which is wrong. Always divide a term by the previous one. Always verify r by checking at least two consecutive ratios: term2/term1 and term3/term2 should be the same.

Before applying a formula, write down a, r, and n (if finite). Verify r by checking multiple ratios. This setup prevents careless errors.

Series 1: 1+2+4+8+...+256 (finite). a=1, r=2. 256=2^8, so n=9 terms. S_9=1(1-2⁹)/(1-2)=(1-512)/(-1)=511. Series 2: 5+10+20+40 (finite). a=5, r=2, n=4. S_4=5(1-2⁴)/(1-2)=5(1-16)/(-1)=5(15)=75. Series 3: 1+1/3+1/9+1/27+... (infinite). a=1, r=1/3. |r|=1/3<1, so it converges. S=1/(1-1/3)=1/(2/3)=3/2. Series 4: 100+50+25+12.5+... (infinite). a=100, r=1/2. S=100/(1-1/2)=100/(1/2)=200. For each series, identify a and r, determine if it is finite or infinite, then apply the appropriate formula.

After calculating, verify by computing the first few terms manually and estimating. For infinite series, check that |r|<1 (otherwise the series diverges). For finite series, verify that the formula gives a result larger than the sum of the first few terms.

Geometric series appear in 0-2 ACT Math questions, often in word problems about compound interest, population growth, or decay. Once you master the formula, you solve these problems without listing all terms, saving significant time. Many students try to calculate each term manually; knowing the formula lets you solve in under one minute, earning points that other students leave on the table.

Spend 15 minutes this week solving 10 geometric series problems (include both finite and infinite, and at least one word problem). Time yourself; each should take 1-2 minutes with the formula. By test day, you will recognize geometric series instantly and apply the formula confidently, unlocking points on questions that seem like they should take much longer.