下面给出了一个数列的前四项。

(i)
完成上表中关于 $T_5$ 的行。
(ii)
用 n 表示 $T_n$ 的表达式。

答
(iii)
求 $T_{88}$ 的值。

答
(iv)
$T_r$ 和 $T_{r+1}$ 是数列中的相邻两项。
用 p 表示 $T_{p+1}-T_p$ 的表达式并化简。

答
(v)
解释为什么该数列中任意两个相邻项的差不能为 4。

答
(vi)
另一个数列的前四项为 8、18、32 和 50。
通过使用 (i) 和 (ii) 或其他方法，写出该数列第 n 项 $T_n$ 的用 n 表示的表达式。

答

探究规律与第n项一次式化简用字母表示数乘法公式去括号与提公因式计算器使用 English

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完整解法

4(i)

1. Extend the pattern to the fifth term and evaluate it in the same table format.

The next odd number after $7$ is $9$ .

$T_5=1\times2+3\times2+5\times2+7\times2+9\times2$

$=2\times(1+3+5+7+9)$

$=2\times25$

$=50$

答案

$T_5=1\times2+3\times2+5\times2+7\times2+9\times2=2\times(1+3+5+7+9)=2\times25=50$

4(ii)

1. Identify the general pattern from the given terms.

From the table,
$T_n=2\times(1+3+5+\cdots+(2n-1)).$
The sum of the first $n$ odd numbers is $n^2$ .
So,
$T_n=2\times n^2=2n^2.$

答案

$T_n=2n^2$

4(iii)

1. Substitute $n=88$ into the nth term formula and evaluate.

From part (ii),
$T_n=2n^2$
So
$T_{88}=2(88)^2=2(7744)=15488$

答案

$T_{88}=15488$

4(iv)

1. Write expressions for the two consecutive terms using the nth term formula.

From part (ii),
$T_n=2n^2.$
So
$T_{p+1}=2(p+1)^2 \quad \text{and} \quad T_p=2p^2.$

2. Form the required difference between the consecutive terms.

$T_{p+1}-T_p=2(p+1)^2-2p^2.$

3. Expand and simplify the expression.

\begin{aligned} T_{p+1}-T_p&=2(p^2+2p+1)-2p^2\\ &=2p^2+4p+2-2p^2\\ &=4p+2. \end{aligned}

答案

$T_{p+1}-T_p=4p+2$

4(v)

1. Use the expression for the difference between consecutive terms and justify why it cannot be 4.

From part (iv),
$T_{p+1}-T_p=4p+2.$
This is always $2$ more than a multiple of $4$ , so it cannot be equal to $4$ .

答案

Since $T_{p+1}-T_p=4p+2$ , the difference is always $2$ more than a multiple of $4$ . Hence it cannot be $4$ .

4(vi)

1. Relate the new sequence to the original sequence.

From earlier, the original sequence has nth term $2n^2$ .

The new sequence $8,18,32,50$ is the original sequence starting from $T_2$ , so its nth term is the original $(n+1)$ th term.

2. Substitute $n+1$ into the original nth term expression.

$T_n=2(n+1)^2$

答案