Find the term of an AP whose term is 38 and term is 73.

Understand the Question The term of an Arithmetic Progression (AP) is given by the formula: where is the first term and is the common difference. We are given two terms: the term () and the term (). We can set up a system of two linear equations in terms of and , solve for both variables, and then calculate the term (). Step 1 · Set up equations for given terms The general term of an AP is For the term: For the term: Step 2 · Find the common difference and first term Subtract equation (1) from equation (2): Substitute into equation (1): Step 3 · Calculate the 31st term Using and in the formula for : Answer > Common Mistakes Formula index error: Using instead of , which incorrectly yields for instead of . Sign error in subtraction: Making a sign mistake when calculating , resulting in instead of .

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Question 1

Find the $31^{\text{st}}$ term of an AP whose $11^{\text{th}}$ term is 38 and $16^{\text{th}}$ term is 73.

Solution

Understand the Question

The $n^{\text{th}}$ term of an Arithmetic Progression (AP) is given by the formula: $a_n = a + (n - 1)d$ where $a$ is the first term and $d$ is the common difference.
We are given two terms: the $11^{\text{th}}$ term ( $a_{11} = 38$ ) and the $16^{\text{th}}$ term ( $a_{16} = 73$ ).
We can set up a system of two linear equations in terms of $a$ and $d$ , solve for both variables, and then calculate the $31^{\text{st}}$ term ( $a_{31} = a + 30d$ ).

Step 1 · Set up equations for given terms

The general term of an AP is $a_n = a + (n - 1)d$

For the $11^{\text{th}}$ term:

\begin{aligned} a_{11} &= a + (11 - 1)d \\ 38 &= a + 10d \quad \dots (1) \end{aligned}

For the $16^{\text{th}}$ term:

\begin{aligned} a_{16} &= a + (16 - 1)d \\ 73 &= a + 15d \quad \dots (2) \end{aligned}

Step 2 · Find the common difference and first term

Subtract equation (1) from equation (2):

\begin{aligned} (a + 15d) - (a + 10d) &= 73 - 38 \\ 5d &= 35 \\ d &= \dfrac{35}{5} \\ d &= 7 \end{aligned}

Substitute $d = 7$ into equation (1):

\begin{aligned} a + 10(7) &= 38 \\ a + 70 &= 38 \\ a &= 38 - 70 \\ a &= -32 \end{aligned}

Step 3 · Calculate the 31st term

Using $a = -32$ and $d = 7$ in the formula for $a_{31}$ :

\begin{aligned} a_{31} &= a + (31 - 1)d \\ &= a + 30d \\ &= -32 + 30(7) \\ &= -32 + 210 \\ &= 178 \end{aligned}

Answer

$178$

Common Mistakes

Formula index error: Using $a_n = a + nd$ instead of $a_n = a + (n - 1)d$ , which incorrectly yields $a + 11d$ for $a_{11}$ instead of $a + 10d$ .
Sign error in subtraction: Making a sign mistake when calculating $a = 38 - 70$ , resulting in $+32$ instead of $-32$ .

Question 1

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