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Read the numbers and decide what the next number should be. 5, 15, 6, 18, 7, 21, 8.

The given problem aims to find the next number that will follow the number series 5, 15, 6, 18, 7, 21, and 8.

The article is based on the concept of Arithmetic Sequence. An Arithmetic Sequence is formulated by adding a fixed constant d in subsequent numbers repeatedly from the starting number a.

The number sequence can be increasing or decreasing at a fixed rate by addition, subtraction, multiplication, or division of a certain constant or factor in the previous number.

Expert Answer

Given that:

$Number$ $Series$ $=$ $5$, $15$, $6$, $18$, $7$, $21$, $8$.

We must find the next number in the given series by using the concept of $Arithmetic$ $Sequence$.

We can identify the next number by 2 methods as mentioned below.

Method-1

The Second, Fourth and Sixth numbers in the sequence are the multiples of 3 of their previous numbers, respectively.

Second Number $15=5\times3$. Thus, the second number is the first number multiplied by $3$.

Fourth Number $18=6\times3$. Thus, the fourth number is the third number multiplied by $3$.

Sixth Number $21=7\times3$. Thus, the sixth number is the fifth number multiplied by $3$.

By continuing this arithmetic sequence, we can calculate that the eighth number of the sequence is the seventh number multiplied by $3$.

We know that the seventh number of the arithmetic sequence is given as $8$.

Hence, the eighth number of the arithmetic sequence will be calculated as follows:

\[Eighth\ Number=Seventh\ Number\times3\]

\[Eighth\ Number=8\times3\]

\[Eighth\ Number=24\]

Thus, the next number (eighth number) in the given arithmetic sequence is $24$.

Method-2

Let:

$A1=5$

$B1=15$

$A2=6$

$B2=18$

$A3=7$

$B3=21$

$A4=8$

$B4=? $

By Considering $A1$ and $B1$, we assess that:

\[\frac{B1}{A1}=\frac{15}{5}\]

\[B1=3\times\ A1\]

By Considering $A2$ and $B2$, we assess that:

\[\frac{B2}{A2}=\frac{18}{6}\]

\[B2=3\times\ A2\]

By Considering $A3$ and $B3$, we assess that:

\[\frac{B3}{A3}=\frac{21}{7}\]

\[B3=3\times\ A3\]

Now that we know $A4=8$, by using the above-mentioned pattern of multiplication, we get:

\[B4=3\times\ A4\]

\[B4=3\times8\]

\[B4=24\]

So the next number $B4$ in the given arithmetic sequence is $24$.

Numerical Result

The next number in the given arithmetic sequence $5$, $15$, $6$, $18$, $7$, $21$, $8$ will be $24$.

Example

Find the number that comes next in the given $Arithmetic$ $series$: $8$, $6$, $9$, $23$, $87? $.

Solution

To find the next number in the given arithmetic sequence, we need to find the pattern or relation based on which the subsequent numbers are increasing or decreasing.

$A=8$

$B=6$

$C=9$

$D=23$

$E=87$

$F=? $

We will express the number $B$ in terms of the number $A$:

\[B=(A\times1)-2\]

\[6=(8\times1)-2\]

We will express the number $C$ in terms of the number $B$:

\[C=(B\times2)-3\]

\[9=(6\times2)-3\]

We will express the number $D$ in terms of the number $C$:

\[D=(C\times3)-4\]

\[23=(9\times3)-4\]

We will express the number $E$ in terms of the number $D$:

\[E=(D\times4)-5\]

\[87=(23\times4)-5\]

So to find the next number $F$ in the sequence, we will use the above relation with the incremental constants.

\[F=(E\times5)-6\]

\[F=(87\times5)-6\]

\[F=429\]

So our required next number in the series is $429$.

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