Let's solve the problem step by step:

The problem involves an arithmetic sequence where:

• The $5^{\text{th}}$ term ($a_5$) is $30$,
• The $20^{\text{th}}$ term ($a_{20}$) is $75$.

We need to find:

a) The common difference $d$:

The general formula for the $n^{\text{th}}$ term of an arithmetic sequence is:

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

For the $5^{\text{th}}$ term: 30 = a_1 + 4d \tag{1}

For the $20^{\text{th}}$ term: 75 = a_1 + 19d \tag{2}

Subtract Equation (1) from Equation (2): $(75 - 30) = (a_1 + 19d) - (a_1 + 4d)$ $45 = 15d$ $d = 3$

b) The first term ($a_1$):

Substitute $d = 3$ into Equation (1): $30 = a_1 + 4(3)$ $30 = a_1 + 12$ $a_1 = 18$

c) What number should be added to the first term to get the $31^{\text{st}}$ term?

Using the same general formula: $a_{31} = a_1 + (31 - 1)d$ $a_{31} = 18 + 30(3)$ $a_{31} = 18 + 90 = 108$

The number to be added to the first term to get the $31^{\text{st}}$ term is: $108 - 18 = 90$

Final Answers:

a) The common difference $d = 3$.
b) The first term $a_1 = 18$.
c) The number to be added is $90$.

Would you like further clarification or a detailed explanation for any step?

Related Questions:

1. What is the $10^{\text{th}}$ term of this sequence?
2. How many terms are there in this sequence if the last term is 300?
3. What is the sum of the first 20 terms in this sequence?
4. Can you express the general term $a_n$ in terms of $n$?
5. If $a_{15} = x$, how can we find $x$?

Tip:

Always subtract equations systematically in arithmetic sequence problems to eliminate unknowns!