5. Arithmetic Progressions

6 topics1 h

Chapter 5 of NCERT Mathematics for Class 10: Arithmetic Progressions. Three sections — What makes a list an AP, Reaching any term and Adding the terms up. Six videos, 1 h in all.

Worked answers to the 49 exercise questions in this chapter: Exercise 5.1 · Exercise 5.2 · Exercise 5.3 · Exercise 5.4 (Optional)

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Part 1

What makes a list an AP

14 minA fixed step between neighbours is the whole definitionAn arithmetic progression is not a list that changes steadily. It is a list whose step from each entry to the next is one unchanging number - which is why a shrinking list counts, a list that never moves counts, and a list that becomes five quarters of itself every time does not.14 min · 13 parts14 minTesting a given list, and what "finite" versus "infinite" changesDeciding whether a list is an arithmetic progression is a test that can only ever fail cleanly. One disagreeing pair of neighbouring differences settles it as no, for ever. Any number of agreeing pairs settles nothing beyond the entries you were shown.14 min · 14 parts

Part 2

Reaching any term

13 minBuilding the rule for the nth term out of repeated additionThe first entry of an arithmetic progression is where you stand before you move, so it costs no steps. Every entry after it costs exactly one. That single sentence is the whole of the nth term rule, and the famous off-by-one is not a wrong number - it is the next entry.13 min · 13 parts16 minWorking backwards from a term to its position, or to a or dThe nth-term rule ties four quantities together in one equation, so any three of them settle the fourth. Solving for the POSITION is the exception: its answer has to be a whole number AND at least one, and those are two separate demands that catch two separate mistakes.16 min · 14 parts

Part 3

Adding the terms up

13 minGauss's pairing trick, and why it generalisesGauss's trick is not a fact about the numbers 1 to 100. Write the total out, write it out again backwards, and stack the two rows: one row rises by exactly what the other falls by, so every column comes to the same amount - and n copies of one number is a multiplication rather than n additions.13 min · 14 parts16 minTwo versions of the total, and choosing between themThere are not two formulas for the total of an arithmetic progression. The last entry is already sitting inside the bracket of the first one, so naming it gives the second - and which of the two you can actually run is settled by the three quantities the question handed you, never by taste.16 min · 15 parts

What you will be able to do

  • State what has to be true of a list of numbers for it to be called an arithmetic progression, in terms of the difference between neighbouring entries
  • Apply the AP test to a given list by comparing every available pair of neighbouring differences, not just the first pair
  • Extend an AP one step at a time and describe why that method is impractical for a distant entry
  • Rearrange the nth-term rule to make any one of the four quantities the subject
  • Reproduce Gauss's argument for the whole numbers from 1 to 100 and explain what each of its two lines does
  • Show that the two printed totals are the same statement, by substituting the nth-term expression for the last entry

What this chapter assumes you already know

  • Adding and subtracting integers, decimals and fractions, including subtracting a larger number from a smaller one
  • Reading a subscripted symbol such as a₃ as "the third one of these"
  • Substituting a number for a letter in an expression
  • That multiplying by a fixed factor and adding a fixed amount are different operations — carried over from percentage and compound-interest work in earlier classes
  • Subtracting signed numbers, fractions and decimals
  • Simplifying surds far enough to see that √8 is 2√2 and √18 is 3√2
  • Recognising repeated multiplication (a fixed percentage, a fixed factor) as a different rule from repeated addition
  • Collecting repeated addition into multiplication: five lots of 500 is 5 × 500

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "AP means the numbers get bigger."
  • "A fixed rate of increase makes an AP."
  • "Subtract the smaller from the larger."
  • "Zero cannot be a common difference."
  • "Knowing the first term tells you the AP."
  • "The squares 1, 4, 9, 16 are an AP because they follow a clear pattern."
  • "a + 2d is the second term."
  • "Check the first two terms and you are done."