Q55.

Find the arithmetic means in each sequence.

3, ? , ? , ? , ?, ? , 27

Verified

Answer

The arithmetic means in the sequence are 7, 11, 15, 19, 23.

1Step 1. Given Information.

Given there are five arithmetic means between 3 and 27.

2Step 2. Calculation .

The n^th term of an arithmetic sequence is given by $a_{n} = a_{1} + (n - 1) d$ $a_{n} = a_{1} + (n - 1) d$ .

Since there are five arithmetic means between 3 and 27,

The first term is $a_{1} = 3$

And the seventh term is $a_{7} = 27$

Plugging the values in the formula:

$\begin{array}{l} a_{n} = a_{1} + (n - 1) d \\ a_{7} = a_{1} + (7 - 1) d \\ 27 = 3 + 6 d \\ 6 d = 27 - 3 \\ 6 d = 24 \\ d = 4 \end{array}$

Hence the common difference is 4.

Since the first term in the arithmetic sequence is 3:

The next term in the arithmetic sequence is $3 + 4 = 7$

The next term in the arithmetic sequence is $7 + 4 = 11$

The next term in the arithmetic sequence is $11 + 4 = 15$

The next term in the arithmetic sequence is $15 + 4 = 19$

The next term in the arithmetic sequence is $19 + 4 = 23$

And the fourth term in the arithmetic sequence is 27.

3Step 3. Conclusion .

Hence, the arithmetic means in the sequence are 7, 11, 15, 19, 23.