Question Video: Finding the Number of Terms in a Given Arithmetic Sequence given the Sum of All Terms Mathematics

Find the number of terms in which the sum of the sequence (160, 144, 128, οΌΏ) is 0.


Video Transcript

Find the number of terms in which the sum of the sequence 160, 144, 128, blank is zero.

Since our sum is finite and it’s not continuous forever, we will find this sum by using the partial sum formula: 𝑛 divided by two times two π‘Ž plus 𝑛 minus one times 𝑑, where π‘Ž is the first term and 𝑑 is the common difference between all of the terms. So π‘Ž would be 160 and the difference between 160 and 144 would be negative 16. And the difference between 144 and 128 is also negative 16. And it said that we wanna find this when the sum is equal to zero.

So we can plug that in for 𝑆 sub 𝑛. So 𝑛 is the number that we’re solving for. So now that we’ve plugged in, let’s evaluate. Two times 160 is 320 and 𝑛 minus one times negative 16 is negative 16𝑛 plus 16. 320 plus 16 is 336. And now we distributed and divided by two to both terms. And now we can take out a greatest common factor of eight 𝑛 cause we can take eight out of both terms and 𝑛 out of both terms.

Now to solve, we need to set each factor equal to zero. So we set eight 𝑛 equal to zero and 21 minus 𝑛 equal to zero. Dividing by eight gives us that 𝑛 equals zero and adding 𝑛 to the right-hand side over here gives us that 21 is equal to 𝑛. So it would make sense that if we would add zero numbers together, we’re gonna get a sum of zero. However, we know that this is not the case, we have 160, 144, and 128 already. So 𝑛 would have to be 21.

So the number of terms so- for which the sum of this sequence would equal zero would have to be 21 terms.

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