Arithmetic Progressions • Topic 5 of 5

Special Results and Word Problems Based on AP

Special Results (Sums of Special Sequences):

1. Sum of first n natural numbers: $ 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2} $ This is an AP with $ a = 1, d = 1 $.

2. Sum of first n even numbers: $ 2 + 4 + 6 + \dots + 2n = n(n+1) $ This is an AP with $ a = 2, d = 2 $.

3. Sum of first n odd numbers: $ 1 + 3 + 5 + \dots + (2n-1) = n^2 $ This is an AP with $ a = 1, d = 2 $.

Word Problem Strategies:

Read the problem carefully and identify what forms the AP.
Find the first term $ a $ and common difference $ d $.
Determine whether the problem asks for a particular term ($ a_n $) or a sum ($ S_n $).
Translate words like "each year increase by fixed amount" → $ d $
"after n years" → look for $ a_n $ or $ S_n $
"total after n days" → usually $ S_n $

Consecutive integer problems: Three consecutive terms in AP can be taken as: $ a-d, a, a+d $ Four consecutive terms: $ a-3d, a-d, a+d, a+3d $ This simplifies algebra.

SPECIAL SUM FORMULAE (Visual Proof)
   
   SUM OF FIRST n NATURAL NUMBERS
   1 + 2 + 3 + ... + n = n(n+1)/2
   
   Example: n=4 → 1+2+3+4 = 10
   Formula: 4×5/2 = 10 ✓
   
   SUM OF FIRST n EVEN NUMBERS
   2 + 4 + 6 + ... + 2n = n(n+1)
   
   Example: n=4 → 2+4+6+8 = 20
   Formula: 4×5 = 20 ✓
   
   SUM OF FIRST n ODD NUMBERS
   1 + 3 + 5 + ... + (2n-1) = n²
   
   Example: n=4 → 1+3+5+7 = 16
   Formula: 4² = 16 ✓

STRATEGY FOR WORD PROBLEMS
   
   ┌─────────────────────────────────────┐
   │ PROBLEM → Identify a and d          │
   │         ↓                           │
   │   Find term?  Use a_n = a+(n-1)d    │
   │   Find sum?   Use S_n formula       │
   │   Find n?     Solve quadratic       │
   └─────────────────────────────────────┘

TAKING CONSECUTIVE TERMS IN AP
   
   3 terms:  a-d,  a,  a+d   (sum = 3a)
   4 terms:  a-3d, a-d, a+d, a+3d (sum = 4a)
   5 terms:  a-2d, a-d, a, a+d, a+2d (sum = 5a)

Solved Examples

Worked Example

Find the sum of all odd numbers between 0 and 50.

Solution

Odd numbers: 1, 3, 5, …, 49
This is AP with $ a = 1, d = 2 $
Find n: $ a_n = 1 + (n-1)2 = 49 $ → $ 1 + 2n - 2 = 49 $ → $ 2n - 1 = 49 $ → $ 2n = 50 $ → $ n = 25 $
Sum = $ n^2 = 25^2 = 625 $ (using sum of first n odd numbers formula)

Answer: 625.

Worked Example

Find three numbers in AP whose sum is 24 and product is 440.

Solution

Let numbers be $ a-d, a, a+d $
Sum: $ (a-d) + a + (a+d) = 3a = 24 $ → $ a = 8 $
Product: $ (a-d)×a×(a+d) = a(a^2 - d^2) = 8(64 - d^2) = 440 $
$ 64 - d^2 = 55 $ → $ d^2 = 9 $ → $ d = 3 $ or $ d = -3 $
Numbers: 5, 8, 11 or 11, 8, 5

Answer: 5, 8, 11 (or 11, 8, 5).

Worked Example

A ladder has rungs 25 cm apart. The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and bottom rungs are 2.5 m apart, find the total length of wood used for the rungs.

Solution

Distance between top and bottom rungs = 2.5 m = 250 cm
Gap between rungs = 25 cm
Number of gaps = $ 250/25 = 10 $ gaps → number of rungs = $ 10 + 1 = 11 $
Lengths: 45, ?, ?, …, 25 (AP with 11 terms)
$ a = 45, l = 25, n = 11 $
Total length = $ S_{11} = \frac{n}{2}(a + l) = \frac{11}{2}(45 + 25) = \frac{11}{2}×70 = 11×35 = 385 $ cm

Answer: 385 cm of wood.

Key Points

Sum of first n natural numbers: $ \frac{n(n+1)}{2} $
Sum of first n even numbers: $ n(n+1) $
Sum of first n odd numbers: $ n^2 $
For word problems, identify $ a, d, n $ from the situation.
Use $ a-d, a, a+d $ for three consecutive terms in AP to simplify algebra.
Check if the problem asks for a term ($ a_n $) or a sum ($ S_n $).

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.The sum of the first $n$ odd natural numbers is:

Explanation: $1+3+\dots=n^2$.

Q2.The sum of the first $n$ even natural numbers is:

Explanation: $2+4+\dots=n(n+1)$.

Q3.The number of integers from $1$ to $100$ divisible by $5$ is:

Explanation: $5,10,\dots,100$.

Q4.If the angles of a triangle are in AP, the middle angle is:

Explanation: Sum $180^\circ$, middle $=60^\circ$.

Practice more MCQs → 📝 Assignment · 35 marks