Problem relating to summation of arithmetic sequence

Algebra Level 2

Let there be an arithmetic sequence where the first term is − 5 -5 and the difference is 2 2 . The sum of the first k k terms of the sequence is bigger than 27 27 and smaller than 55 55 . k ∈ N k\in \mathbb{N} . Find the value of k k .


The answer is 10.

Nir S. by Nir S. , 20, Israel

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1 solution

Nir S.
Aug 1, 2019

a n a_n = − 5 + 2 ( n − 1 ) = − 5 + 2 n − 2 = 2 n − 7 -5+2(n-1) = -5+2n-2=2n-7 , and therefore a k a_k = 2 k − 7 2k-7 .

The sum of the first k k terms of the series can be calculated with the arithmetic series summation formula: S k S_k = k ( a 1 + a k ) 2 \frac{k\left(a_1+a_k\right)}{2} = k ( − 5 + 2 k − 7 ) 2 = k ( 2 k − 12 ) 2 = k ( k − 6 ) = k 2 − 6 k \frac{k\left(-5+2k-7\right)}{2}=\frac{k\left(2k-12\right)}{2}=k\left(k-6\right)=k^2-6k

We know that 27 < S k < 55 27<S_k<55 and therefore 27 < k 2 − 6 k < 55 27<k^2-6k<55 , which is a simple inequality to solve and is equivalent to: − 5 < k < − 3 o r 9 < k < 11 -5<k<-3\quad \mathrm{or}\quad \:9<k<11 . We know that k k is a natural number since it represents a number of terms in a series, and it's also written in the body of the question. The only natural number within the range we found is 10 10 , and thus k = 10 k=10

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