4.2 Q-13

Exercise 4.2 — Question 13

Problem Statement

This question involves finding specific terms or properties of an arithmetic sequence using the general term formula and properties of arithmetic progressions.

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Key Concepts Required

Arithmetic Sequence

A sequence $a_1,a_2,a_3,\dots$ is arithmetic if the difference between consecutive terms is constant: $d=a_n-a_{n-1}=\text{constant}$

General (nth) Term

$a_n=a_1+(n-1)d$ where:

• $a_1$ = first term
• $d$ = common difference
• $n$ = term number

Finding $d$ from Two Terms

If the $m^{th}$ term is $a_m$ and the $n^{th}$ term is $a_n$: $d=\frac{a_n-a_m}{n-m}$

Finding Number of Terms

If $a_1$, $a_n$, and $d$ are known: $n=\frac{a_n-a_1}{d}+1$

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Worked Approach

Step 1: Identify the given information (first term $a_1$, common difference $d$, or specific terms).

Step 2: Write the general term formula $a_n=a_1+(n-1)d$.

Step 3: Substitute known values and solve for the unknown.

Step 4: Verify by checking that the difference between consecutive terms is constant.

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Example

Find the 15th term of the arithmetic sequence: $3,7,11,15,\dots$

Solution:

• $a_1=3$, $d=7-3=4$
• $a_{15}=3+(15-1)(4)=3+56=59$

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