The Life Starter
Sign up
Grade 11 · lesson 3 of 4

Pair First and Last Terms in an Arithmetic Sum

Terms equally far from opposite ends of an arithmetic sequence have equal pair sums.

About 25 minutes, plus any extended practice

Download the grade pack

Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: Find the sum of the four-term arithmetic sequence 2,4,6,8. Candidate answers: (A) 19 (B) 20 (C) 21 (D) 40.
  • Independent check inputs: Find the sum of the four-term arithmetic sequence 3,5,7,9. Candidate answers: (A) 23 (B) 24 (C) 25 (D) 48.

Learn the idea

A sequence distinguishes an indexed term from the sum of several terms. Arithmetic and geometric structures support different formulas, while a recursive definition determines each next term from previously defined values. Terms equally far from opposite ends of an arithmetic sequence have equal pair sums. Pairing gives the sum formula as number of terms times the average of first and last.

Here is the method on the supplied teaching case. Pair first with last and second with third; each pair sums to 10. Two pairs total 20. Equivalently S=4(first+last)/2.

Use the stated number of terms, including both endpoints. The average of endpoints equals the mean for an arithmetic sequence, but that shortcut need not work for an arbitrary list.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: Find the sum of the four-term arithmetic sequence 2,4,6,8. Candidate answers: (A) 19 (B) 20 (C) 21 (D) 40. Solution: Pair first with last and second with third; each pair sums to 10. Two pairs total 20. Equivalently S=4(first+last)/2. The final answer is 20. Use the stated number of terms, including both endpoints. The average of endpoints equals the mean for an arithmetic sequence, but that shortcut need not work for an arbitrary list.

Your turn

Check your reasoning

Your answer is temporary and is not saved to an account. Download your current answer before leaving or reloading. Written work on paper or in another app stays in your own file.

Helpful tips
  • Terms equally far from opposite ends of an arithmetic sequence have equal pair sums.
  • Use the stated number of terms, including both endpoints.
Find the sum of the four-term arithmetic sequence 3,5,7,9.

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

24

Pair first with last and second with third; each pair sums to 12. Two pairs total 24. Equivalently S=4(first+last)/2.

Take it further

Changed-input task: Find the sum of the four-term arithmetic sequence 9,11,13,15. Candidate answers: (A) 47 (B) 48 (C) 49 (D) 96. Show the mathematical reason, not only the selected answer. Expected result: 48. Pair first with last and second with third; each pair sums to 24. Two pairs total 48. Equivalently S=4(first+last)/2.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.

More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. Find the sum of the four-term arithmetic sequence 4,6,8,10.
Need a hint?
  • Terms equally far from opposite ends of an arithmetic sequence have equal pair sums.
  • Use the stated number of terms, including both endpoints.
2. Find the sum of the four-term arithmetic sequence 5,7,9,11.
Need a hint?
  • Terms equally far from opposite ends of an arithmetic sequence have equal pair sums.
  • Use the stated number of terms, including both endpoints.
3. Find the sum of the four-term arithmetic sequence 6,8,10,12.
Need a hint?
  • Terms equally far from opposite ends of an arithmetic sequence have equal pair sums.
  • Use the stated number of terms, including both endpoints.

Your choices are temporary. Feedback appears after you check all the answers.

Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.

Check the subject sources

These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.