Sequences and SeriesIntroduction

Make your choice in your head, then test it. Drag the day. The blue bar is everything Offer A has paid so far, and the purple bar is everything Offer B has paid. Find the first day on which Offer B is ahead.

Day: ${d}

Most people pick Offer A, and for three whole weeks they look clever: on day 21, Offer B is still ₹28.49 behind. Then the doubling takes over. By day 30, Offer A has paid ₹30,000 — and Offer B has paid more than ₹1 crore.

Look at what you were really watching. Each offer is a list of numbers in a fixed order — a first amount, a second amount, a third, and so on: rupees and paise. A list like that is called a sequence. The bars showed something else: what you get by adding up the terms of a sequence. That sum is called a series.

Sequences that follow a fixed pattern are called progressions. In Class 10 you met the arithmetic progression, where you add the same number every time — like Offer A's running total, . This chapter is about its much faster cousin, the geometric progression, where you multiply by the same number every time — like Offer B. You will find its terms and its sums, meet the geometric mean, and discover a rule that ties it to the arithmetic mean.

A 4000-Year-Old Pattern

In India, Aryabhata (born 476 CE) gave rules for the sums of the squares and of the cubes of the natural numbers in his Aryabhatiya, written around 499 CE, together with a rule for adding up an arithmetic sequence that starts at any term. Brahmagupta (born 598), Mahavira (around 850) and Bhaskara II (1114–1185) also studied such sums. The Italian mathematician Leonardo Fibonacci (about 1170–1250) described a sequence that you will meet in a few minutes. In 1671 the Scottish mathematician James Gregory used the words infinite series, and only much later, once algebra and set theory were made rigorous, did sequences and series get the careful definitions we use today.

Drag these milestones into order, earliest at the top.

Four thousand years on, the two big questions are still the same: what comes next? and what does it all add up to? This chapter answers both, starting with the first.