Fill in the blanks in the following table, given that is the first term, the common difference and the term of the AP:

Understand the Question The term of an Arithmetic Progression (AP) with first term and common difference is given by the formula: Among the four variables (, , , ), if any three are known, the fourth can be calculated by substituting the given values into this formula. Part (i) Given , , , find . Step 1 · Calculate the term Given , , and . Using the formula : Answer > (i) Part (ii) Given , , , find . Step 1 · Calculate the common difference Given , , and . Using the formula : Answer > (ii) Part (iii) Given , , , find . Step 1 · Calculate the first term Given , , and . Using the formula : Answer > (iii) Part (iv) Given , , , find . Step 1 · Calculate the number of terms Given , , and . Using the formula : Answer > (iv) Part (v) Given , , , find . Step 1 · Calculate the term Given , , and . Using the formula : Answer > (v) Common Mistakes Sign Errors with Negative Numbers: In row (iii), when multiplying , students often forget the negative sign, writing instead of . Forgetting to Solve for : In row (iv), stopping at instead of adding to obtain . Constant Sequence (): In row (v), calculating complicated values when ; if common difference is , every term remains equal to .

Arithmetic Progressions | Exercise 5.2

Question 1

Fill in the blanks in the following table, given that $a$ is the first term, $d$ the common difference and $a_n$ the $n^{\text{th}}$ term of the AP:

Solution

Understand the Question

The $n^{\text{th}}$ term $a_n$ of an Arithmetic Progression (AP) with first term $a$ and common difference $d$ is given by the formula:

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

Among the four variables ( $a$ , $d$ , $n$ , $a_n$ ), if any three are known, the fourth can be calculated by substituting the given values into this formula.

(i) Given $a = 7$ , $d = 3$ , $n = 8$ , find $a_n$ .

Step 1 · Calculate the $n^{\text{th}}$ term

Given $a = 7$ , $d = 3$ , and $n = 8$ .

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

\begin{aligned} a_8 &= 7 + (8 - 1) \times 3 \\ &= 7 + 7 \times 3 \\ &= 7 + 21 \\ &= 28 \end{aligned}

Answer

(i) $a_n = 28$

(ii) Given $a = -18$ , $n = 10$ , $a_n = 0$ , find $d$ .

Step 1 · Calculate the common difference

Given $a = -18$ , $n = 10$ , and $a_n = 0$ .

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

\begin{aligned} 0 &= -18 + (10 - 1)d \\ 0 &= -18 + 9d \\ 9d &= 18 \\[0.6em] d &= \dfrac{18}{9} = 2 \end{aligned}

Answer

(ii) $d = 2$

(iii) Given $d = -3$ , $n = 18$ , $a_n = -5$ , find $a$ .

Step 1 · Calculate the first term

Given $d = -3$ , $n = 18$ , and $a_n = -5$ .

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

\begin{aligned} -5 &= a + (18 - 1) \times (-3) \\ -5 &= a + 17 \times (-3) \\ -5 &= a - 51 \\ a &= -5 + 51 \\ a &= 46 \end{aligned}

Answer

(iii) $a = 46$

(iv) Given $a = -18.9$ , $d = 2.5$ , $a_n = 3.6$ , find $n$ .

Step 1 · Calculate the number of terms

Given $a = -18.9$ , $d = 2.5$ , and $a_n = 3.6$ .

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

\begin{aligned} 3.6 &= -18.9 + (n - 1) \times 2.5 \\ 3.6 + 18.9 &= (n - 1) \times 2.5 \\ 22.5 &= (n - 1) \times 2.5 \\[0.6em] n - 1 &= \dfrac{22.5}{2.5} \\[0.6em] n - 1 &= 9 \\ n &= 9 + 1 = 10 \end{aligned}

Answer

(iv) $n = 10$

(v) Given $a = 3.5$ , $d = 0$ , $n = 105$ , find $a_n$ .

Step 1 · Calculate the $n^{\text{th}}$ term

Given $a = 3.5$ , $d = 0$ , and $n = 105$ .

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

\begin{aligned} a_{105} &= 3.5 + (105 - 1) \times 0 \\ &= 3.5 + 104 \times 0 \\ &= 3.5 + 0 \\ &= 3.5 \end{aligned}

Answer

(v) $a_n = 3.5$

Common Mistakes

Sign Errors with Negative Numbers: In row (iii), when multiplying $17 \times (-3)$ , students often forget the negative sign, writing $+51$ instead of $-51$ .
Forgetting to Solve for $n$ : In row (iv), stopping at $n - 1 = 9$ instead of adding $1$ to obtain $n = 10$ .
Constant Sequence ( $d = 0$ ): In row (v), calculating complicated values when $d = 0$ ; if common difference is $0$ , every term remains equal to $a$ .

Question 1

More questions in Exercise 5.2