Select the right expression for the following sum and prove that your statement is true for all natural numbers n. $$p _ { n } : 8 + 6 + 4 \ldots ( 10 - 2 n )$$
A) $9 n - 2 n ^ { 2 }$
B) $11 n - n ^ { 2 }$
C) $\quad 9 n - n ^ { 2 }$
D) $\quad 11 n - 2 n ^ { 2 }$
E) none of the above

Question

Quizplus · Accepted Answer

We can see that the terms in the sum are decreasing by 2 each time, so we have an arithmetic sequence. The first term is 8 and the common difference is -2. We want to find the sum of the first n terms.

The formula for the sum of n terms of an arithmetic sequence is:

$$S_n = \frac{n}{2}(a_1+a_n)$$

where $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the nth term.

We can find $a_n$ by substituting $n$ into the expression for the nth term of the sequence:

$$a_n = 10 - 2n$$

We can now substitute the values we have into the formula for the sum of an arithmetic sequence:

$$S_n = \frac{n}{2}(a_1+a_n)$$

$$S_n = \frac{n}{2}(8 + (10 - 2n))$$

$$S_n = \frac{n}{2}(18 - 2n)$$

$$S_n = 9n - n^2$$

Therefore the correct expression is (C), and it holds true for all natural numbers n.

Select the Right Expression for the Following Sum and Prove pn:8+6+4…(10−2n)p _ { n } : 8 + 6 + 4 \ldots ( 10 - 2 n )pn​:8+6+4…(10−2n)

Select the Right Expression for the Following Sum and Prove $p _ { n } : 8 + 6 + 4 \ldots ( 10 - 2 n )$