6. A sequence is defined so that $t_n = a/(n-1)d$. If $a=3, d=5$, find $t_n$ and $t_4+ t_5+ t_6+ t_7$ is $31$ and $t_4+t_5+t_6+t_7+t_8=16$. Find the value of $t_4$ (if possible).

Key takeaway: This question appears to mix an arithmetic-sequence formula with two sums that do not match the same sequence. The first step is to interpret the intended rule carefully, then check consistency.

Step 1: Read the intended formula

The standard arithmetic-sequence formula is

$t_n = a + (n-1)d$

With $a=3$ and $d=5$,

$t_n = 3 + (n-1)5 = 5n - 2$

Step 2: Find $t_4$ from this formula

$t_4 = 5(4) - 2 = 18$

So if the sequence is arithmetic with first term $3$ and common difference $5$, then

$\boxed{t_4 = 18}$

Step 3: Check the given sums

Using this same sequence,

$t_4+t_5+t_6+t_7 = 18+23+28+33 = 102$

and

$t_4+t_5+t_6+t_7+t_8 = 18+23+28+33+38 = 140$

These do not match the values $31$ and $16$ stated in the problem.

Conclusion

The data in the problem are inconsistent. Under the usual arithmetic-sequence interpretation, $t_4=18$, but the given sums cannot both be true for that sequence.

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Pitfalls the pros know 👇 Do not force inconsistent numbers to fit a formula. When a problem statement gives values that contradict the sequence rule, the correct response is to identify the inconsistency rather than invent a value.

What if the problem changes? If the intended formula was something other than an arithmetic sequence, then $t_n$ would need to be redefined before solving. With only the information shown, the most reasonable interpretation is $t_n=3+5(n-1)$, which gives $t_4=18$.

Tags: arithmetic sequence, common difference, consistency check