Answer Any 2 Questions From 16 To 18.16. The Algebraic Expression For The Sum Of N Terms Of An Arithmetic Sequence Is N 2 + N N^2 + N N 2 + N .(a) Find The First Term Of This Arithmetic Sequence.(b) Find The Sum Of The First 10 Terms Of This Arithmetic Sequence.

Mar 8, 2025 by ADMIN 263 views

**Answering Questions on Arithmetic Sequences**

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference. The sum of the first n terms of an arithmetic sequence can be calculated using the formula: $S_n = \frac{n}{2} [2a + (n-1)d]$ , where $S_n$ is the sum of the first n terms, a is the first term, and d is the common difference.

Question (a): Find the first term of this arithmetic sequence.

Given that the algebraic expression for the sum of n terms of an arithmetic sequence is $n^2 + n$ , we can equate this expression to the formula for the sum of an arithmetic sequence: $n^2 + n = \frac{n}{2} [2a + (n-1)d]$ . To find the first term, we need to simplify the equation and solve for a.

Step 1: Simplify the equation

We can start by multiplying both sides of the equation by 2 to eliminate the fraction: $2n^2 + 2n = n [2a + (n-1)d]$ .

Step 2: Expand the right-hand side

Expanding the right-hand side of the equation, we get: $2n^2 + 2n = 2an + nd - nd$ .

Step 3: Simplify the equation

Simplifying the equation, we get: $2n^2 + 2n = 2an + nd$ .

Step 4: Equate coefficients

Equating the coefficients of n on both sides of the equation, we get: $2 = 2a$ .

Step 5: Solve for a

Solving for a, we get: $a = \frac{2}{2} = 1$ .

Therefore, the first term of the arithmetic sequence is 1.

Question (b): Find the sum of the first 10 terms of this arithmetic sequence.

Now that we have found the first term, we can use the formula for the sum of an arithmetic sequence to find the sum of the first 10 terms. We are given that the algebraic expression for the sum of n terms of an arithmetic sequence is $n^2 + n$ . We can substitute n = 10 into this expression to find the sum of the first 10 terms.

Step 1: Substitute n = 10

Substituting n = 10 into the expression, we get: $10^2 + 10 = 100 + 10$ .

Step 2: Simplify the expression

Simplifying the expression, we get: $110$ .

Therefore, the sum of the first 10 terms of the arithmetic sequence is 110.

Conclusion

Frequently Asked Questions

Q: What is an arithmetic sequence?

A: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference.

Q: What is the formula for the sum of an arithmetic sequence?

A: The formula for the sum of an arithmetic sequence is: $S_n = \frac{n}{2} [2a + (n-1)d]$ , where $S_n$ is the sum of the first n terms, a is the first term, and d is the common difference.

Q: How do I find the first term of an arithmetic sequence?

A: To find the first term of an arithmetic sequence, you can use the formula for the sum of an arithmetic sequence and the given algebraic expression for the sum of n terms. You can equate the two expressions and solve for a.

Q: How do I find the sum of the first n terms of an arithmetic sequence?

A: To find the sum of the first n terms of an arithmetic sequence, you can use the formula for the sum of an arithmetic sequence: $S_n = \frac{n}{2} [2a + (n-1)d]$ .

Q: What is the difference between an arithmetic sequence and a geometric sequence?

A: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant, while a geometric sequence is a sequence of numbers in which the ratio between any two consecutive terms is constant.

Q: Can I use the formula for the sum of an arithmetic sequence to find the sum of a geometric sequence?

A: No, the formula for the sum of an arithmetic sequence cannot be used to find the sum of a geometric sequence. You need to use a different formula for the sum of a geometric sequence.

Q: How do I find the common difference of an arithmetic sequence?

A: To find the common difference of an arithmetic sequence, you can use the formula for the sum of an arithmetic sequence and the given algebraic expression for the sum of n terms. You can equate the two expressions and solve for d.

Q: Can I use the formula for the sum of an arithmetic sequence to find the sum of a finite arithmetic sequence?

A: Yes, you can use the formula for the sum of an arithmetic sequence to find the sum of a finite arithmetic sequence.

Q: Can I use the formula for the sum of an arithmetic sequence to find the sum of an infinite arithmetic sequence?

A: No, you cannot use the formula for the sum of an arithmetic sequence to find the sum of an infinite arithmetic sequence. You need to use a different formula for the sum of an infinite arithmetic sequence.

Q: What is the formula for the sum of an infinite arithmetic sequence?

A: The formula for the sum of an infinite arithmetic sequence is: $S = \frac{a}{1 - r}$ , where $S$ is the sum of the infinite sequence, $a$ is the first term, and $r$ is the common ratio.

Q: How do I find the sum of an infinite arithmetic sequence?

A: To find the sum of an infinite arithmetic sequence, you can use the formula for the sum of an infinite arithmetic sequence: $S = \frac{a}{1 - r}$ .

Q: Can I use the formula for the sum of an arithmetic sequence to find the sum of a finite geometric sequence?

A: No, you cannot use the formula for the sum of an arithmetic sequence to find the sum of a finite geometric sequence. You need to use a different formula for the sum of a finite geometric sequence.

Q: What is the formula for the sum of a finite geometric sequence?

A: The formula for the sum of a finite geometric sequence is: $S_n = \frac{a(1 - r^n)}{1 - r}$ , where $S_n$ is the sum of the first n terms, $a$ is the first term, and $r$ is the common ratio.

Q: How do I find the sum of a finite geometric sequence?

A: To find the sum of a finite geometric sequence, you can use the formula for the sum of a finite geometric sequence: $S_n = \frac{a(1 - r^n)}{1 - r}$ .

Q: Can I use the formula for the sum of an arithmetic sequence to find the sum of a geometric sequence with a common ratio of 1?

A: No, you cannot use the formula for the sum of an arithmetic sequence to find the sum of a geometric sequence with a common ratio of 1. You need to use a different formula for the sum of a geometric sequence with a common ratio of 1.

Q: What is the formula for the sum of a geometric sequence with a common ratio of 1?

A: The formula for the sum of a geometric sequence with a common ratio of 1 is: $S_n = na$ , where $S_n$ is the sum of the first n terms, and $a$ is the first term.

Q: How do I find the sum of a geometric sequence with a common ratio of 1?

A: To find the sum of a geometric sequence with a common ratio of 1, you can use the formula for the sum of a geometric sequence with a common ratio of 1: $S_n = na$ .

Q: Can I use the formula for the sum of an arithmetic sequence to find the sum of a finite arithmetic sequence with a common difference of 0?

A: No, you cannot use the formula for the sum of an arithmetic sequence to find the sum of a finite arithmetic sequence with a common difference of 0. You need to use a different formula for the sum of a finite arithmetic sequence with a common difference of 0.

Q: What is the formula for the sum of a finite arithmetic sequence with a common difference of 0?

A: The formula for the sum of a finite arithmetic sequence with a common difference of 0 is: $S_n = na$ , where $S_n$ is the sum of the first n terms, and $a$ is the first term.

Q: How do I find the sum of a finite arithmetic sequence with a common difference of 0?

A: To find the sum of a finite arithmetic sequence with a common difference of 0, you can use the formula for the sum of a finite arithmetic sequence with a common difference of 0: $S_n = na$ .

Q: Can I use the formula for the sum of an arithmetic sequence to find the sum of an infinite arithmetic sequence with a common difference of 0?

A: No, you cannot use the formula for the sum of an arithmetic sequence to find the sum of an infinite arithmetic sequence with a common difference of 0. You need to use a different formula for the sum of an infinite arithmetic sequence with a common difference of 0.

Q: What is the formula for the sum of an infinite arithmetic sequence with a common difference of 0?

A: The formula for the sum of an infinite arithmetic sequence with a common difference of 0 is: $S = a$ , where $S$ is the sum of the infinite sequence, and $a$ is the first term.

Q: How do I find the sum of an infinite arithmetic sequence with a common difference of 0?

A: To find the sum of an infinite arithmetic sequence with a common difference of 0, you can use the formula for the sum of an infinite arithmetic sequence with a common difference of 0: $S = a$ .

Q: Can I use the formula for the sum of an arithmetic sequence to find the sum of a finite geometric sequence with a common ratio of 1?

A: No, you cannot use the formula for the sum of an arithmetic sequence to find the sum of a finite geometric sequence with a common ratio of 1. You need to use a different formula for the sum of a finite geometric sequence with a common ratio of 1.

Q: What is the formula for the sum of a finite geometric sequence with a common ratio of 1?

A: The formula for the sum of a finite geometric sequence with a common ratio of 1 is: $S_n = na$ , where $S_n$ is the sum of the first n terms, and $a$ is the first term.

Q: How do I find the sum of a finite geometric sequence with a common ratio of 1?

A: To find the sum of a finite geometric sequence with a common ratio of 1, you can use the formula for the sum of a finite geometric sequence with a common ratio of 1: $S_n = na$ .

Q: Can I use the formula for the sum of an arithmetic sequence to find the sum of an infinite geometric sequence with a common ratio of 1?

A: No, you cannot use the formula for the sum of an arithmetic sequence to find the sum of an infinite geometric sequence with a common ratio of 1. You need to use a different formula for the sum of an infinite geometric sequence with a common ratio of 1.

Q: What is the formula for the sum of an infinite geometric sequence with a common ratio of 1?

A: The formula for the sum of an infinite geometric sequence with a common ratio of 1 is: $S = a$ , where $S$ is the sum of the infinite sequence, and $a$ is the first term.

Q: How do I find the sum of an infinite geometric sequence with a common ratio of 1?

A: To find the sum of an infinite geometric sequence with a common ratio of 1, you can use the formula for the sum of an infinite geometric sequence with a common ratio of 1: $S = a$ .