2024 AMC 10A Problems/Problem 19
- The following problem is from both the 2024 AMC 10A #19 and 2024 AMC 12A #12, so both problems redirect to this page.
Contents
Problem
The first three terms of a geometric sequence are the integers 
and 
where 
What is the sum of the digits of the least possible value of 
Solution 1
For a geometric sequence, we have 
, and we can test values for 
. We find that 
and 
works, and we can test multiples of 
in between the two values. Finding that none of the multiples of 5 divide 
besides 
itself, we know that the answer is 
.
(Note: To find the value of without bashing, we can observe that 
, and that multiplying it by 
gives us 
, which is really close to . ~ YTH)
(Note: The reason why 
is because 
. Rearranging this gives 
)
~eevee9406
Solution 2
We have 
. We want to find factors 
and 
where 
such that 
is minimized, as 
will then be the least possible value of . After experimenting, we see this is achieved when 
and 
, which means our value of is 
, so our sum is .
~i_am_suk_at_math_2
See also
| 2024 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 18 |
Followed by Problem 20 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
| 2024 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 11 |
Followed by Problem 13 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
