Day 5 — Sequences And Series
Time: ~30 min · Topic: sequences and series
The problem
The first three terms of an arithmetic sequence are 5, 11, 17. What is the sum of the first 15 terms of this sequence?
Hint: Find the common difference first, then look up the formula for the sum of an arithmetic series: S = n/2 * (2a + (n-1)d)
Try it yourself
Take a few minutes. Don't peek at the steps. If you get stuck, the hint above is enough to nudge you forward.
Step-by-step solution
- Find the common difference. d = 11 - 5 = 6
- Identify the key values. First term a = 5, common difference d = 6, number of terms n = 15
- Apply the arithmetic series sum formula. S = n/2 * (2a + (n-1)d) = 15/2 * (2(5) + 14(6)) = 15/2 * (10 + 84) = 15/2 * 94
- Calculate. S = 15 * 47 = 705
Answer: The sum of the first 15 terms is 705.
Why this matters
The young Carl Friedrich Gauss famously discovered the arithmetic series formula at age 10 when his teacher told the class to add up all numbers from 1 to 100 — he got 5050 in seconds while his classmates were still adding!
This day in math: The Man Who Tamed Randomness
April 25, 1903 · Andrey Kolmogorov
On April 25, 1903, Andrey Kolmogorov was born in Tambov, Russia. Orphaned as an infant and raised by his aunt, he went on to become one of the most influential mathematicians of the 20th century. In 1933, he published his groundbreaking axioms of probability theory, finally putting the mathematics of chance on a rigorous foundation after centuries of informal use.
Why it matters: Kolmogorov's axioms are the bedrock of modern statistics, machine learning, and artificial intelligence. Every time an algorithm predicts the weather, recommends a video, or diagnoses a disease, it's built on the framework he created.
Did you know?
The Fibonacci sequence secretly doubles as a miles-to-kilometers converter! Take any Fibonacci number as miles, and the NEXT Fibonacci number gives you a shockingly accurate conversion to kilometers. 5 miles ≈ 8 km. 8 miles ≈ 13 km. 13 miles ≈ 21 km. It works every time.
The ratio between consecutive Fibonacci numbers approaches the golden ratio, approximately 1.618. This is remarkably close to the actual miles-to-kilometers conversion factor of 1.609. So each pair of consecutive Fibonacci numbers — like (5, 8), (8, 13), or (13, 21) — naturally mirrors the mile-to-kilometer relationship with less than 0.6% error.
Done?
- Solved the problem (or read through the steps)
- Read the history note
- Read the "did you know" fact