Question 18

Explanation

Given the arithmetic sequence $5, 8, 11, \ldots$.

From the sequence:

• First term $a = 5$
• Common difference $b = 8 - 5 = 3$

Formula for the $n$-th term of an arithmetic sequence:

Formula for the sum of the first $n$ terms:

Evaluating Statement (1)

Statement: The value of $a$ is greater than the value of $b$.

Therefore, statement $(1)$ is true.

Evaluating Statement (2)

Statement: $U_{11} = a \times 7$.

Calculate $U_{11}$:

Calculate $a \times 7$:

Since $U_{11} = 35$ and $a \times 7 = 35$, then $U_{11} = a \times 7$.

Therefore, statement $(2)$ is true.

Evaluating Statement (3)

Statement: $S_6 = b \times 25$.

Calculate $S_6$:

Calculate $b \times 25$:

Since $S_6 = 75$ and $b \times 25 = 75$, then $S_6 = b \times 25$.

Therefore, statement $(3)$ is true.

Evaluating Statement (4)

Statement: $U_{18} = S_5$.

Calculate $U_{18}$:

Calculate $S_5$:

Since $U_{18} = 56$ and $S_5 = 55$, then $U_{18} \neq S_5$.

Therefore, statement $(4)$ is false.