The 1st term of an artihmetic sequence is 8 and the common difference is 3. Find the sum of \(a_5+a_6+a_7+\cdots+a_{99}+a_{100}\).

ShockFish Feb 3, 2022

#1**+7 **

The 5th term is 20. The pattern is the n-th term is 3(n-1) + 8, so the 100-th term of the pattern is 3(99) + 8 or 305. Thus, all the numbers added from 20 to 305 would be added together, with a difference of 3 between each number. Since we know there are a total of 96 numbers added together, we can subtract 17 from each number, then divide each number by 3. Then the numbers from 1 - 96 would be added together, which equals 4656.

Then we multiply 4656 by 3 to get 13968. Then since we subtracted 17 from 96 numbers, we have to add (17 x 96) to 13968. That gets 15600. **Thus, the answer is 15600.**

Correct me if I got it wrong :D

proyaop Feb 3, 2022

#2**+4 **

Use the formula for the sum of an arithmetic sequence:

S==96/2 * [2* 20 + (3*95)], solve for S

S ==48 * [40 + 285]

**S ==48 * 325 ==15,600**

Guest Feb 3, 2022