Manveetha wrote:

niks18 wrote:

SajjadAhmad wrote:

What is the one-thousandth term in sequence S ?

(1) The fifth term in S is 47.

(2) Each term in S following the first term is generated by multiplying the preceding term by 4 and adding 1.

The correct answer is (C). The rule for constructing the sequence is given

in (2). And the point of reference is given in (1). There is no need to try to find

the value of the thousandth term. You need only recognize that it is possible

to do so.

To find \(T_{1000}\)

Statement 1: \(T_5=47\). but no relation about other terms is given. Hence

InsufficientStatement 2: implies, \(T_n=4T_{n-1}+1\). But value of no term is given. Hence

InsufficientCombining 1 & 2: We know the relation between terms and value of one term. Hence each term in the sequence can be calculated.

SufficientOption

CCan you explain what I'm missing? Do we not need to know the first term?

Each term in S following the first term is generated by multiplying the preceding term by 4 and adding 1.

How can we use the relation between terms if we don't know T4, T3, T2 and most importantly T1?

Hi

ManveethaFrom Statement 2 we know that \(T_n=4T_{n-1}+1\)

so \(T_2=4*T_1+1\). Similarly

\(T_3=4T_2+1 = 4(4*T_1+1)+1\)

\(T_4=4T_3+1= 4*[4(4*T_1+1)+1]+1\)

\(T_5=4T_4+1=4*[4*[4(4*T_1+1)+1]+1]+1\).

Now you know the value of \(T_5\), so substitute it in the last equation to get the value of \(T_1\). Similarly you can \(T_{1000}\) in terms of \(T_1\) and hence the final answer.

As this is a DS question we do not need to actually calculate the value but only need to know that the value can be calculated and only a unique value is possible.