In a G. P., the last term is 1024 and the common ratio is \(2 .\) Its 20 th term from the end is (A) \([1 /(512)]\) (B) \([1 /(1024)]\) (C) \([1 /(256)]\) (D) 512

The last term in a geometric progression (\textbf{GP}) is an essential aspect for solving problems that deal with the sequence's endpoint. In relation to our exercise, the last term was given as 1024. To understand how we might use the last term, let's define a GP as a sequence of numbers where each term after the first is found by multiplying the previous one by a constant called the common ratio. Now, knowing the last term, we can perform backwards calculations to find any other term preceding it, using the formula for the last (or Nth) term of a GP which is given by:
\[T_N = a \times r^{(N-1)}\]
where \(T_N\) is the last term, \(a\) is the first term, \(r\) is the common ratio, and \(N\) is the total number of terms. This formula becomes a cornerstone of other calculations within the GP.

The common ratio of a GP is arguably its defining characteristic. In our example, the common ratio is given as 2, which means every term in the sequence is twice the previous term. Mathematically, for any GP, the common ratio \(r\) is found by dividing any term by its preceding term, i.e.,
\[r = \frac{T_{n+1}}{T_n}\]
where \(T_{n+1}\) is a term in the sequence and \(T_n\) is the preceding term. The concept of the common ratio is crucial for various reasons; it allows us to navigate both forwards and backwards within the GP, and it's also integral to deriving the general formula for the nth term (which we'll see in the next section), as well as understanding the series' behavior – whether it's converging, diverging, or oscillating.

Deriving the nth term of a geometric progression is a fundamental skill in dealing with GPs. The nth term formula is an expression that, given the first term and the common ratio, provides the value of any term within the sequence. The formula is written as:
\[T_n = a \times r^{(n-1)}\]
where \(T_n\) is the nth term, \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

Tracking terms from the End

When we're interested in a term from the end of a GP, like the 20th term from the last term in our original exercise, we count backwards. For example, if a GP has \(N\) terms, the nth term from the end is the \((N-n+1)\)th term from the beginning. This adjustment allows us to leverage the known last term and the common ratio to pinpoint the value of any term from the end. Remember, for the nth term formula to be useful in such contexts, we must have a solid grasp of the sequence's total length, or be clever with the provided information to deduce that term's value.