Finding Arithmetic Sequence Terms

Pre-Calculus · Axiom Academy

EXAMPLE Finding Arithmetic Sequence Terms Three worked problems using the explicit formula a_n = a_1 + (n-1)d Find the 20th term of an arithmetic sequence where a_1 = 5 and d = 3 . Each step of 3 adds one more unit of the common difference; the 20th term lands at 62. An arithmetic sequence has a_1 = 100 and a_ 10 = 73 . Find the common difference d . The line slopes downward: dropping 27 over 9 steps means each step falls by 3. Finding Which Term Equals a Value In the sequence , which term equals 97 ? The dashed target line crosses the sequence exactly at term 19. Nice work — you handled all three directions of the same formula. One formula, three jobs: a_n = a_1 + (n-1)d connects the four quantities a_1 , d , n , and a_n . Know any three, find the fourth. Find a specific term: substitute n and evaluate directly — here a_ 20 = 5 + 19(3) = 62 . Find the common difference: substitute what you know and solve for d — a downward slope gave d = -3 . Find which term hits a value: set the formula equal to that value and solve for n — the target 97 was the 19th term. Because each term changes by the same step d , an arithmetic sequence is just a line in disguise — every question above is really "read this line."

This is the written version of the interactive lesson above. See the full Pre-Calculus course.