Question

Hind the sum of the following arithmetic series: 4+10+16+22+............. +82

Answer 1

Sum = 602

Explanation:

Formula for the sum of an arithmetic series:

The formula for the sum of an arithmetic sequence is given by:

`S_(n)=n/2(a_(1)+a_(n))`, where:

• Sn is the sum,
• n is the number of terms,
• a1 is the first term,
• and an is the last term.

Formula for the nth term of an arithmetic series:

We can determine how many terms the series has using the formula for the nth term of an arithmetic series, which is given by:

`a_(n)=a_(1)+(n-1)d`, where

• an is the nth term,
• a1 is the first term,
• n is the term position (e.g., 1st, 5th, etc.),
• and d is the common difference.

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Step 1: Find the common difference for the nth term formula:

We can first find the common difference (d) by subtracting two consecutive terms (e.g., 10 and 4):

d = 10 - 4

d = 6

Thus, the common difference is 6.

Step 2: Find the term position of 82:

• Knowing the term position of 82 will tell us how many terms the arithmetic series has.

Now, we can find the term position (n) of 82- by substituting 82 for an, 4 for a1, and 6 for d in the nth term formula:

`(82=4+(n-1)*6)/-4\\\\(78=(n-1)*6)/6\\\\(13=(n-1))+1\\\\14=n`

Since 82 is the 14th term, the arithmetic series has 14 terms in all.

Step 3: Find the sum of the arithmetic series:

Now, we can find the sum (Sn) of the arithmetic series by substituting 14 for n, 4 for a1, and 82 for an in the arithmetic series sum formula:

`S_(n)=14/2(4+82)\\\\S{n}=7(86)\\\\S{n}=602`

Therefore, the sum of the arithmetic series is 602.