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Question

What is the next number in the series 0,6,24,60,120,210,?


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Solution

Step 1. Difference between the first and second terms in the series.

The given series is 0,6,24,60,120,210,.

Take the difference between the first and second terms:

6-0=624-6=1860-24=36120-60=60210-120=90

Thus, the new series of differences is 6,18,36,60,90.

Step 2: Difference between the first and second terms of the new series6,18,36,60,90.

Consider the sixth term of the series is X.

6,18,36,60,90,X

18-6=1236-18=1860-36=2490-60=30

Hence, X-90=Y

Thus, the new series of differences is 12,18,24,30,Y...(ii).

Step 3: Difference between the first and second terms of the new series12,18,24,30,Y.

Where the fifth term of the series is Y.

12,18,24,30,Y

18-12=624-18=630-24=6

Hence, Y-30=6

Thus, the difference is 6, which is constant.

Step 4. Find the next term of the given series.

Add 6 to the fourth term of the series of 12,18,24,30 in Step 2 to find the value of Y:

Y=30+6Y=36

Add 36 to the fifth term of the series 6,18,36,60,90 of Step 1 to find the value of X:

X=90+36X=126

Now, add 126 to the sixth term of the given series:

210+126=336

Hence, the next term in the given series is 336 and the series is 0,6,24,60,120,210,360


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