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Question: What is the next term to this series \(2,3,7,16,32\) and \(57,...\)? A)\(94\) B)\(93\) C)\(92\) D)...

What is the next term to this series 2,3,7,16,322,3,7,16,32 and 57,...57,...?
A)9494 B)9393 C)9292 D)9595

Explanation

Solution

We have to look at the difference of the numbers to notice the pattern which the series follows. Here, the difference is 1,4,9,16,251,4,9,16,25 … which are the squares of the given numbers. So the next difference will be 3636 adding it to the last number given in the series will give you the answer.

Complete step-by-step answer:
The given series is 2,3,7,16,322,3,7,16,32 and 57,...57,...
We have to find the next term in the series. We have to look at the difference of the numbers to notice the pattern which the series follows. On observing the pattern of the series we see that on adding 1,4,9,16,251,4,9,16,25 … respectively in the series (which are the squares of the natural numbers) we are getting the next term,
⇒2+(1)2=2+1=3 3+(2)2=3+4=7 7+(3)2=7+9=16 16+(4)2=16+16=32 32+(5)2=32+25=57    \Rightarrow 2 + {\left( 1 \right)^2} = 2 + 1 = 3 \\\ 3 + {\left( 2 \right)^2} = 3 + 4 = 7 \\\ 7 + {\left( 3 \right)^2} = 7 + 9 = 16 \\\ 16 + {\left( 4 \right)^2} = 16 + 16 = 32 \\\ 32 + {\left( 5 \right)^2} = 32 + 25 = 57 \\\ {\text{ }} \\\
So for the next term we'll add the square of 66 to the last term given in the series.
⇒57+(6)2=57+36=93\Rightarrow 57 + {\left( 6 \right)^2} = 57 + 36 = 93
So the correct answer is ‘B’.

Note: We can also solve this by formula. Since here the difference is the in pattern of the square of the natural numbers, the formula for the nth{{\text{n}}^{{\text{th}}}} term is-
⇒\Rightarrow nth{{\text{n}}^{{\text{th}}}} Term=2n3−3n2+n + 126\dfrac{{{\text{2}}{{\text{n}}^3} - 3{{\text{n}}^2} + {\text{n + 12}}}}{6}
Here we have to find 7th{7^{{\text{th}}}} term so on putting the value in formula we get-
⇒7thterm = 2(7)3−3(7)2+7+126\Rightarrow {7^{{\text{th}}}}{\text{term = }}\dfrac{{2{{\left( 7 \right)}^3} - 3{{\left( 7 \right)}^2} + 7 + 12}}{6}
On simplifying the equation, we get,
⇒7thterm = 2(343)−3(49)+196=686−147+196=5586\Rightarrow {7^{{\text{th}}}}{\text{term = }}\dfrac{{2\left( {343} \right) - 3\left( {49} \right) + 19}}{6} = \dfrac{{686 - 147 + 19}}{6} = \dfrac{{558}}{6}
On dividing the number, we get
⇒7thterm = 5586=93\Rightarrow {7^{{\text{th}}}}{\text{term = }}\dfrac{{558}}{6} = 93
So we will get the same answer.