The terms of a sequence of partial sums are defined by Sₙ = ∑ⁿ...

Question

The terms of a sequence of partial sums are defined by Sₙ = ∑ⁿₖ₌₁ k² , for n=1, 2, 3, .....Evaluate the first four terms of the sequence.

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says let use of N be equal to the sum of M equal to 1 ton of the quantity of m + 1 in quantity squared 4 N equal to 1234, and so forth. What are the 1st 4 terms of this sequence. So to find the 1st 4 terms of our sequence use of N. So that means we're looking at the cases when N is equal to 123, and 4. So, we're just going to calculate those values directly using the formula for use of N that we're given in the problem. So, for the first term, we're going to need N equal to 1, so we'll calculate U1, and this is going to be equal to the sum of M equal to 1 to 1 of the quantity of M plus 1 in quantity squared. So for the sum, we're just going to have a single term with M equal to 1. So evaluating the sum, this is going to be equal to the quantity of 1 plus 1 in quantity squared, and evaluating this expression, we see that U1 is going to be equal to 4. So that is going to be the first term in our sequence. Now for the second term, we're going to need use of n when n is equal to 2. So we'll calculate U2, which is going to be equal to the sum of M equal to 1 to 2 of the quantity of m plus 1 in quantity squared. So performance summation, this is going to be equal to, and here we'll have two terms, one for M equal to 1 and one for M equal to 2. So this is going to be equal to the quantity of 1 plus 1 in quantity squared, plus the quantity of 2 + 1 in quantity squared. Simplifying this expression, this is going to be equal to 4 + 9, which is equal to 13. So that means that our second term, U2 is equal to 13. Now, for the 3rd term, we'll need to calculate U of N, when N is equal to 3. So we'll calculate U3. NU3 is going to be equal to the sum of M equal to 1 to 3 of the quantity of M plus 1 in quantity squared. So performing in the sum, this is going to be equal to the quantity of 1 plus 1 in quantity squared, plus the quantity of 2 + 1 in quantity squared, plus the quantity of 3 plus 1 in quantity squared. Simplify this expression, this is going to be equal to 4 + 9 + 16. And evaluating this expression, this is going to be equal to 29. So that means that U3 is equal to 29. And for the fourth term in our sequence that we need to calculate, we'll need to calculate use of N when N is equal to 4, so we'll calculate U4, and this is going to be equal to the sum of M equal to 1 to 4 of the quantity of M plus 1 in quantity squared. So performing the sum again we'll have 4 terms in our sum. So this is going to be equal to the quantity of 1 plus 1 in quantity squared, plus the quantity of 2 plus 1 in quantity squared, plus the quantity of 3 plus 1 in quantity squared, plus the quantity of 4 plus 1 in quantity squared. So finding this expression, this is going to be equal to 4 + 9 + 16 + 25. And evaluating this expression, this is going to be equal to 54. So that means that U4 is going to be equal to 54. So just to recap the values that we found, we found that U1 is equal to 4. We found that U2 is equal to 13. We found that U3 was equal to U29, and we found, we found that U4 was equal to 54. So those are the answers that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

The terms of a sequence of partial sums are defined by Sₙ = ∑ⁿₖ₌₁ k² , for n=1, 2, 3, .....Evaluate the first four terms of the sequence.

Key Concepts

Sequence of Partial Sums

Summation Notation and Formula for Squares

Evaluating Terms of a Sequence

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