Exercise 4.2 — Question 17

Problem Statement

Find the arithmetic sequence in which the $7^{th}$ term is $38$ and the $13^{th}$ term is $74$. Also find the sum of the first $20$ terms.

Solution

Step 1: Set Up Equations Using the General Term Formula

The general term of an arithmetic sequence is: $a_n=a_1+(n-1)d$

Using the given information:

• $7^{th}$ term: $a_7=a_1+6d=38$     ...(i)
• $13^{th}$ term: $a_{13}=a_1+12d=74$     ...(ii)

Step 2: Solve for the Common Difference $d$

Subtract equation (i) from equation (ii): $(a_1+12d)-(a_1+6d)=74-38$ $6d=36$ $d=6$

Step 3: Solve for the First Term $a_1$

Substitute $d=6$ into equation (i): $a_1+6(6)=38$ $a_1+36=38$ $a_1=2$

Step 4: Write the Arithmetic Sequence

With $a_1=2$ and $d=6$, the sequence is: $2,8,14,20,26,32,38,\dots$

Step 5: Find the Sum of the First 20 Terms

The formula for the sum of the first $n$ terms of an arithmetic series is: $S_n=\frac{n}{2}[2a_1+(n-1)d]$

Substituting $n=20$, $a_1=2$, $d=6$: $S_{20}=\frac{20}{2}[2(2)+(20-1)(6)]$ $S_{20}=10[4+114]$ $S_{20}=10\times 118$ $\boxed{S_{20}=1180}$

Key Formulas Used