这是一个数列的前四项。 2 6 10 14 求该数列第 $n$ 项关于 $n$ 的表达式。 答案 ... 该数列中有一项是 82。
求这一项对应的 $n$ 值。 答案 ... 解释为什么 360 不是该数列中的项。 答案 ... ... 另一个数列的前 $n$ 项和为 $-4n^2+54n$。
求该数列的第 8 项。 答案 ...

1. Find the common difference and form the nth term expression.

The sequence is $2, 6, 10, 14, \ldots$

The common difference is $6-2=4$.

So the nth term is of the form $4n+c$.

When $n=1$, $4(1)+c=2$, so $c=-2$.

Hence the nth term is $4n-2$.

1. Set the nth term equal to 82 and solve for $n$.

From part (a), the nth term is $4n-2$.

$4n-2=82$

$4n=84$

$n=21$

1. Check whether 360 can be written as a term of the sequence.

From part (a), the $n$th term is $4n-2$.

If $360$ is a term,
$4n-2=360$
$4n=362$
$n=90.5$
Since $n$ must be a positive integer, $360$ is not a term of the sequence.

1. Find the sums of the first 8 terms and the first 7 terms.

Given $S_n=-4n^2+54n$.

$S_8=-4(8)^2+54(8)=-256+432=176$

$S_7=-4(7)^2+54(7)=-196+378=182$

2. Subtract consecutive sums to find the 8th term.

The 8th term is

$T_8=S_8-S_7$

$T_8=176-182=-6$