For what value of , are the terms of two APs: and equal?

Understand the Question The term of an Arithmetic Progression (AP) with first term and common difference is given by the formula: We find the expression for the term of each AP. Since the terms of both APs are equal, we equate them and solve the linear equation for . Step 1 · Find the Term of the First AP For the first AP: First term, Common difference, Using Step 2 · Find the Term of the Second AP For the second AP: First term, Common difference, Using Step 3 · Equate the Terms and Solve for Equating Answer > Common Mistakes Distributive Property Error: Forgetting to multiply both terms in by (e.g., writing as instead of ). Sign Errors during Transposition: Making errors when rearranging terms across the equals sign, such as writing instead of . Non-integer Value of : The term number must always be a positive integer (). A fraction or negative value indicates a calculation mistake.

Arithmetic Progressions | Exercise 5.2

Question 15

For what value of $n$ , are the $n^{\text{th}}$ terms of two APs: $63, 65, 67, \dots$ and $3, 10, 17, \dots$ equal?

Solution

Understand the Question

The $n^{\text{th}}$ term of an Arithmetic Progression (AP) with first term $a$ and common difference $d$ is given by the formula: $a_n = a + (n - 1)d$
We find the expression for the $n^{\text{th}}$ term of each AP.
Since the $n^{\text{th}}$ terms of both APs are equal, we equate them and solve the linear equation for $n$ .

Step 1 · Find the $n^{\text{th}}$ Term of the First AP

For the first AP: $63, 65, 67, \dots$

First term, $a = 63$
Common difference, $d = 65 - 63 = 2$

Using $a_n = a + (n - 1)d$

\begin{aligned} a_n &= 63 + (n - 1)2 \\ &= 63 + 2n - 2 \\ &= 2n + 61 \end{aligned}

Step 2 · Find the $n^{\text{th}}$ Term of the Second AP

For the second AP: $3, 10, 17, \dots$

First term, $b = 3$
Common difference, $d' = 10 - 3 = 7$

Using $b_n = b + (n - 1)d'$

\begin{aligned} b_n &= 3 + (n - 1)7 \\ &= 3 + 7n - 7 \\ &= 7n - 4 \end{aligned}

Step 3 · Equate the $n^{\text{th}}$ Terms and Solve for $n$

Equating $a_n = b_n$

\begin{aligned} 2n + 61 &= 7n - 4 \\ 61 + 4 &= 7n - 2n \\ 65 &= 5n \\[0.6em] n &= \dfrac{65}{5} \\ n &= 13 \end{aligned}

Answer

$13$

Common Mistakes

Distributive Property Error: Forgetting to multiply both terms in $(n-1)$ by $d$ (e.g., writing $(n-1)7$ as $7n - 1$ instead of $7n - 7$ ).
Sign Errors during Transposition: Making errors when rearranging terms across the equals sign, such as writing $61 - 4$ instead of $61 + 4$ .
Non-integer Value of $n$ : The term number $n$ must always be a positive integer ( $n \in \mathbb{N}$ ). A fraction or negative value indicates a calculation mistake.

Question 15

More questions in Exercise 5.2