Question

In another case , p and 2p are the first and second terms of an arithmetic progression. The nth term is 336 and the sum of the first n terms is 7224. Write down two equations in n and p and hence find the value of n and p.

Answer 1

Hint:
With the given terms we can find the common difference and using the nth term formula ${a_n} = a + (n - 1)d$ we can find a equation substituting in the formula of ${S_n} = \dfrac{n}{2}\left[ {2a + (n - 1)d} \right]$. we can find the value of n and p.

Complete step by step solution:
We are given that the first two terms of an AP are p and 2p
$ \Rightarrow a = p,{a_2} = 2p$
The common difference is given by subtracting the first term from the second term
$ \Rightarrow $ 2p – p = p
The nth of the sequence is given by
$ \Rightarrow {a_n} = a + (n - 1)d$
Here we have the nth term to be 336
$
   \Rightarrow 336 = p + (n - 1)p \\
   \Rightarrow 336 = p + pn - p = pn \\
   \Rightarrow np = 336 \\
 $
We know that the sum of the first n terms is given by
$ \Rightarrow {S_n} = \dfrac{n}{2}\left[ {2a + (n - 1)d} \right]$
$
   \Rightarrow 7224 = \dfrac{n}{2}\left[ {2p + (n - 1)p} \right] \\
   \Rightarrow 7224 = \dfrac{n}{2}\left[ {2p + np - p} \right] \\
   \Rightarrow 7224 = \dfrac{n}{2}\left[ {p + np} \right] \\
    \\
 $
Substituting the value of np we get
$
   \Rightarrow 7224 = \dfrac{n}{2}\left[ {p + 336} \right] \\
   \Rightarrow 7224 = \dfrac{{np}}{2} + \dfrac{{336n}}{2} \\
   \Rightarrow 7224 = \dfrac{{336}}{2} + 168n \\
   \Rightarrow 7224 = 168 + 168n \\
   \Rightarrow 7224 = 168(1 + n) \\
   \Rightarrow \dfrac{{7224}}{{168}} = n + 1 \\
   \Rightarrow 43 = n + 1 \\
   \Rightarrow n = 42 \\
 $
Substituting this in np = 336
$
   \Rightarrow 42p = 336 \\
   \Rightarrow p = \dfrac{{336}}{{42}} = 8 \\
 $

Therefore the value of p = 8 and n = 42

Note:
1) In an Arithmetic Sequence the difference between one term and the next is a constant.
2) We can find the common difference of an AP by finding the difference between any two adjacent terms.
3) If we know the initial term, the following terms are related to it by repeated addition of the common difference.