Basic Math and Pre-Algebra - Number Series - Quick Guides

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Study GuideBasic Math and Pre-Algebra–Number Series1. ArithmeƟc Progressions1.1What Is an ArithmeƟc Progression?Anarithmetic progression (AP)is a sequence of numbers where the difference betweenconsecutive terms is always the same.In simple words:•You keep adding (or subtracting) thesame numbereach time.•That fixed number is called thecommon difference.To check if a sequence is an AP:Subtract any term from the next one.If the result is always the same, it’s an arithmetic progression.Example 1:Finding the Next TermQuestion:What is the next number in the progression4, 7, 10, 13, ... ?Step 1: Find the common difference7−4 = 310−7 = 313−10 = 3The difference is always3.Step 2: Add the common difference to the last term13 + 3 =16So, the next number is16.

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Study GuideExample 2:Finding the 40th TermQuestion:What is the 40th term in the progression5, 10, 15, 20, ... ?Step 1: Find the common difference10−5 = 5So, the common difference is5.Step 2: Understand the positionWe know:•1st term = 5•2nd term = 10•3rd term = 15•4th term = 20We need the40th term.Notice that the 40th term is36 terms after the 4th term(because 40−4 = 36).Step 3: Calculate how much the value increases in 36 stepsEach step increases by 5.So,5 × 36 = 180Step 4: Add this to the 4th term20 + 180 =200Therefore, the 40th term is200.

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Study GuideSummary•Anarithmetic progression (AP)is a sequence where the difference between consecutiveterms is constant.•This fixed value is called thecommon difference.•To find the next term, simply add the common difference to the last term.•To find a later term (like the 40th term), calculate how many steps forward you need to go andmultiply that by the common difference.•Then add the result to a known term.2. Geometric ProgressionsWhat Is a Geometric Progression?Ageometric progression (GP)is a sequence of numbers where each term is multiplied by the samenumber to get the next term.Instead of adding the same number (like in an arithmetic progression), here wemultiplyby the samevalue each time.This fixed multiplying value is called thecommon ratio.To check if a sequence is a GP:Divide any term by the one before it.If the answer is always the same, that number is thecommon ratio.Example 1:Finding the Next TermQuestion:What is the next number in the progression1, 2, 4, 8, 16, ... ?Step 1: Find the common ratio2 ÷ 1 = 24 ÷ 2 = 2

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