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How many poles will be in a pile of telephone poles if there are $30$ in the bottom layer, $29$ in the second, on so on, until there is one in the top layer?

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Last updated date: 16th Jun 2024
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Answer
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Hint: The given distribution of the telephone poles tells us that there are a total of $30$ telephone poles in which the number of telephone poles is one in the topmost layer to thirty in the bottommost layer. Therefore, the total number of telephone poles will be obtained as a sum of all the natural numbers from one to thirty. It can be found by using the formula for the sum of first $n$ natural numbers.

Complete step-by-step solution:
According to the question, we have a pile of multiple layers of telephone poles, in which $30$ telephone poles are in the bottom layer, $29$ telephone poles are there in the second layer, and this way we have one telephone in the topmost layer.
From this information, we can say that the number of telephone poles increases as we move from the topmost layer to the bottom layers of the pile. And the increment in the number of telephone poles is equal to one. Therefore, we will have a total of $30$ layers of telephone poles. As the number of telephone poles increases from one to $30$, the total number of telephone poles is given by
$n=1+2+3+........30$.
So the total number of telephone poles is obtained as the sum of first $30$ natural numbers. We know that the sum of first $n$ natural numbers is given by $\dfrac{n\left( n+1 \right)}{2}$. Therefore, the above sum, or the number of telephone poles will be given by
$n=\dfrac{30\left( 30+1 \right)}{2}$
$\Rightarrow n=\dfrac{30\times 31}{2}$
On solving, we finally get
$n=465$

Hence, the total number of telephone poles in the given pile is equal to $465$.

Note:
We must not try to perform the addition of all the thirty natural numbers, as it will be time consuming and also involve huge chances of errors. We must always solve these types of problems using the summation formulae. Since the given series is an AP, so we can also find the sum by using the formula for the sum of an AP, which is given by $S=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]$. An AP or an Arithmetic Progression is a series or sequence where the consecutive terms differ by a common difference.