Question

10. Which is the right answer to the following?
A number sequence has 100 elements. Any of its elements (except for the first and last element) is equal to the product of its neighbors. The product of the first 50
elements, just as the product of all the elements is 27. What is the sum of the first and the second element?​

Answer 1

Step-by-step explanation:

let the series be

(a,ab,b,1/a,1/ab,1/b).. the seq repeats...

for 50 terms:

product=27

product for first 48 * next 2 = 27

1*a^2b=27 ---1

similarly for 100 terms:

1*ab^2=27 ---2

sol 1 and 2

a=3 and b=3

ans 12

Answer 2

The  first and  second element's  sum  is 12.

Step by step explanation:

Given:

A sequence of  numbers. has 100 elements.

Any of sequence's elements (except for the first and last element) is equal to the product of it's neighbors.

The first 50 elements product , just as the product of all  elements is 27.

To find:

The  first and  second element's  sum.

Solution:

Let two numbers of the sequence is p and q.

As given-Any of its elements (except for the first and last element) is equal to the product of its neighbors.

Therefore, The sequence $p,q,\frac{q}{p^{} },\frac{1}{p^{} } , \frac{1}{q^{} } ,\frac{p}{q^{} } ,p,q . . .$

There is repetition the sequence after six terms. The product of first six terms  is 1.

As given-The product of the first 50 elements, just as the product of all the elements is 27.

Product of the  the first 50 elements  $pq=27$  

                                                              $p=\frac{27}{q}$    ----  equation no.01.

( It is because product of other  48 elements  will be 1.)

Product of the  the first 100 elements  $\frac{q^{2} }{p} =27$   ------ equation no.02.

(It is because product of other  96 elements  will be 1.)

Putting the value of p from equation  no.01  in  equation no 02.

$\frac{q^{2} }{(\frac{27}{q} )} =27$

$q^{3} =27\times27=729\\q=\sqrt[3]{729} \\ q =9$

Putting the value of q in equation no.01.

$p=\frac{27}{9}$

$p=3$

Thus,The  first and  second element's  sum  $=p+q$

                                                                                  $=3+9=12$

                                                                                   

Thus,the  first and  second element's  sum  is 12.

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