Question

find the 10th term of the sequnce below
40,__,__,__,__,10

Answer 1

you missed the 5th term the 10 is the 6th term

Answer 2

To find the 10th term of the sequence \(40, \__, \__, \__, \__, 10\), we first must identify a pattern or rule. Given the first term (40) and the last term (10), we can assume that this sequence decreases consistently.

1. **Identifying the common difference**:
- The first term is 40, and the fifth term is 10.
- The number of terms between them, from 1 to 5, is 4 (counting 5 terms: 1 to 5).
- Therefore, the total decrease from 40 to 10 occurs over 4 intervals (the gaps between terms).
- The total decrease is:
\[
40 - 10 = 30
\]
- The common difference (d) is calculated as:
\[
d = \frac{-30}{4} = -7.5
\]

2. **Finding intermediate terms**:
- Now we can find the other terms in the sequence using this common difference.
- The second term:
\[
40 - 7.5 = 32.5
\]
- The third term:
\[
32.5 - 7.5 = 25
\]
- The fourth term:
\[
25 - 7.5 = 17.5
\]
- The fifth term (already known):
\[
17.5 - 7.5 = 10
\]

Putting it all together, the sequence so far is:
\[
40, 32.5, 25, 17.5, 10
\]

Now, to find the 10th term, we need to determine the value of \(a_n\) where \(n = 10\). We can use the formula for the nth term of an arithmetic sequence:
\[
a_n = a_1 + (n-1) \cdot d
\]
where \(a_1 = 40\), \(d = -7.5\) and \(n = 10\):
\[
a_{10} = 40 + (10-1) \cdot (-7.5)
\]
\[
a_{10} = 40 + 9 \cdot (-7.5)
\]
\[
a_{10} = 40 - 67.5
\]
\[
a_{10} = -27.5
\]

Thus, the 10th term of the sequence is \(\boxed{-27.5}\).

Answer 3

I apologize for the oversight. Thank you for your correction. Since 10 is the 6th term of the sequence \(40, \__, \__, \__, \__, 10\), we need to find the 10th term based only on these six terms.

1. **Identify the terms**:
- The first term \(a_1 = 40\)
- The 6th term \(a_6 = 10\)

2. **Calculate the common difference**:
- The difference between the first term and the 6th term is:
\[
40 - 10 = 30
\]
- Since there are 5 gaps between the first term and the 6th term (from \(a_1\) to \(a_6\)), we can find the common difference \(d\) by dividing the total change by the number of intervals (5):
\[
d = \frac{-30}{5} = -6
\]

3. **Find the intermediate terms**:
- Now we can find the values of the intermediate terms:
- 2nd term:
\[
a_2 = 40 + (-6) = 34
\]
- 3rd term:
\[
a_3 = 34 + (-6) = 28
\]
- 4th term:
\[
a_4 = 28 + (-6) = 22
\]
- 5th term:
\[
a_5 = 22 + (-6) = 16
\]
- 6th term (already known):
\[
a_6 = 16 + (-6) = 10
\]

So the sequence now is:
\[
40, 34, 28, 22, 16, 10
\]

4. **Find the 7th, 8th, 9th, and 10th terms**:
- 7th term:
\[
a_7 = 10 + (-6) = 4
\]
- 8th term:
\[
a_8 = 4 + (-6) = -2
\]
- 9th term:
\[
a_9 = -2 + (-6) = -8
\]
- 10th term:
\[
a_{10} = -8 + (-6) = -14
\]

Thus, the 10th term of the sequence is \(\boxed{-14}\).