The sum of the first three terms of a geometric progression is 26, and the sum of their squares is 364. Find the terms of the GP.
Step-by-step solution
Idea: Take the terms as ar, a, ar and let p = r + 1r. Then the sum is a(p + 1) and the sum of squares is a2(p2 − 1) = a2(p + 1)(p − 1). Dividing one by the other removes the hard part.
- Terms ar, a, ar. Sum: a(1r + 1 + r) = 26. Let p = r + 1r: a(p + 1) = 26 … (1).1 mark
- Squares: a2(1r2 + 1 + r2) = 364. Since r2 + 1r2 = p2 − 2, this is a2(p2 − 1) = a2(p + 1)(p − 1) = 364 … (2).1 mark
- (2) ÷ (1): a(p − 1) = 364 ÷ 26 = 14 … (3). (1) − (3): 2a = 12 ⇒ a = 6.1 mark
- From (1): 6(p + 1) = 26 ⇒ p = 103. So r + 1r = 103 ⇒ 3r2 − 10r + 3 = 0 ⇒ (3r − 1)(r − 3) = 0 ⇒ r = 3 or 13.1 mark
- r = 3: terms 2, 6, 18. r = 13: terms 18, 6, 2. The terms of the GP are 2, 6, 18.1 mark
Check: 2 + 6 + 18 = 26 ✓; 4 + 36 + 324 = 364 ✓.
Answer to write in the exam
Let the terms be ar, a, ar; p = r + 1r
a(p + 1) = 26 … (1)
a2(r2 + 1 + 1r2) = a2(p2 − 1) = 364 … (2)
(2) ÷ (1): a(p − 1) = 14 … (3)
(1) − (3): 2a = 12 ⇒ a = 6; p = 103
3r2 − 10r + 3 = 0 ⇒ (3r − 1)(r − 3) = 0 ⇒ r = 3 or 13
∴ Terms: 2, 6, 18 (or 18, 6, 2)
Common mistakes that cost marks
- Squaring the sum (262 = 676) and equating it to 364. The square of a sum is not the sum of the squares.
- Taking the terms as a, ar, ar2 and getting stuck with a degree-4 equation; the symmetric choice ar, a, ar keeps it simple.
- Missing r = 13. It gives the same three numbers in reverse order.
How this can come in the exam
Which three numbers form a GP whose sum is 14?
- 2, 4, 8
- 2, 5, 7
- 1, 4, 9
- 3, 5, 6
Show answer
(A) 2, 4, 8
2, 4, 8 has ratio 2 and sum 14.
Three numbers in GP have sum 21 and the sum of their squares is 189. Find them.
Show answer
a(p + 1) = 21, a2(p2 − 1) = 189 ⇒ a(p − 1) = 9 (1 mark). So a = 6, p = 52 ⇒ 2r2 − 5r + 2 = 0 ⇒ r = 2 or 12 (1 mark). Numbers: 3, 6, 12 (1 mark).Try one yourself
The sum of three numbers in GP is 13 and the sum of their squares is 91. Find the numbers.
Show answer
a(p + 1) = 13, a(p − 1) = 7 ⇒ a = 3, p = 103 ⇒ r = 3 or 13. Numbers: 1, 3, 9.
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