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Q#1 First term $a_1=4$ and common difference $d=5$. Find the formula for the $n^{\text{th}}$ term and find the $6^{\text{th}}$ term of the sequence.



Q#2 Find the missing term '$x$' if $x, 15, 21, 27, \dots$



Q#3 How many terms are there in the arithmetic sequence: $7, \frac{13}{4}, 19, \dots, 61$?



Q#4 If $3, 7, 11, 15, \dots$ is an arithmetic sequence, find the $10^{\text{th}}$ and $12^{\text{th}}$ terms, and check whether 95 is a member of this sequence.



Q#5 Find the product of the following:

A$\left(5 m+3 m n+6 n^2\right)\left(-3 m n+4 n^2\right)$
B$(a+b-c)\left(a^2+b^2+c^2+a c+b c-a b\right)$
C$\left(x^2+y^2+2 x y\right)\left(x^2+y^2-2 x y\right)$

Verified Answer :

  • For the first arithmetic sequence (\(a_1=4, d=5\)): the general term is \(a_n = 5n - 1\), and the 6th term is \(a_6 = 29\).
  • In the sequence \(x, 15, 21, 27\), the constant difference is \(6\), giving the missing term \(x = 9\).
  • The arithmetic sequence \(7, 13, 19, \dots, 61\) consists of \(\boxed{10}\) terms.
  • For the sequence \(3, 7, 11, 15, \dots\), the 10th term is \(\boxed{39}\), the 12th term is \(\boxed{47}\), and \(\boxed{95}\) is the 24th term.
  • The algebraic products are \(\boxed{-9m^2n^2 - 15m^2n - 6mn^3 + 20mn^2 + 24n^4}\), \(\boxed{a^3 + 3abc + b^3 - c^3}\), and \(\boxed{x^4 - 2x^2y^2 + y^4}\).

Explanation

  1. Arithmetic Sequence: First Term and General Formula

To find the \(n^{\text{th}}\) term of an arithmetic sequence, we use the formula \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term and \(d\) is the common difference. Given \(a_1 = 4\) and \(d = 5\), the formula for the \(n^{\text{th}}\) term is:
\[a_n = 4 + (n-1) \times 5 = 4 + 5n - 5 = 5n - 1.\]
To find the \(6^{\text{th}}\) term (\(a_6\)), substitute \(n=6\) into the formula:
\[a_6 = 5(6) - 1 = 30 - 1 = 29.\]

  1. Finding the Missing Term in an Arithmetic Sequence

Given the sequence \(x, 15, 21, 27, \dots\), we identify the constant difference \(d\) by subtracting consecutive terms:
\[d = 21 - 15 = 6.\]
Since this is an arithmetic sequence, the difference between the first two terms must also be \(d\):
\[15 - x = 6 \implies x = 15 - 6 = 9.\]
Thus, the missing term is \(x = 9\).

  1. Counting Terms in an Arithmetic Sequence

In the sequence \(7, 13, 19, \dots, 61\), we have \(a_1 = 7\) and the common difference \(d = 13 - 7 = 6\). The general formula is \(a_n = a_1 + (n-1)d\). Substituting the given values:
\[61 = 7 + (n-1) \times 6.\]
Subtract 7 from both sides:
\[54 = (n-1) \times 6.\]
Divide by 6:
\[9 = n - 1 \implies n = 10.\]
There are 10 terms in the sequence.

  1. Determining Sequence Terms and Membership

For the sequence \(3, 7, 11, 15, \dots\), the first term \(a_1 = 3\) and the common difference \(d = 7 - 3 = 4\). The \(n^{\text{th}}\) term is:
\[a_n = 3 + (n-1) \times 4 = 4n - 1.\]
For the \(10^{\text{th}}\) term:
\[a_{10} = 4(10) - 1 = 39.\]
For the \(12^{\text{th}}\) term:
\[a_{12} = 4(12) - 1 = 47.\]
To check if 95 is a member, set \(a_n = 95\):
\[95 = 4n - 1 \implies 96 = 4n \implies n = 24.\]
Since \(n = 24\) is a positive integer, 95 is indeed the \(24^{\text{th}}\) term of the sequence.

  1. Algebraic Products
A\((5m + 3mn + 6n^2)(-3mn + 4n^2) = -9m^2n^2 - 15m^2n - 6mn^3 + 20mn^2 + 24n^4\).
B\((a + b - c)(a^2 + b^2 + c^2 + ac + bc - ab) = a^3 + 3abc + b^3 - c^3\).
C\((x^2 + y^2 + 2xy)(x^2 + y^2 - 2xy) = (x+y)^2(x-y)^2 = ((x+y)(x-y))^2 = (x^2 - y^2)^2 = x^4 - 2x^2y^2 + y^4\).

Examples

Arithmetic sequences appear in many daily scenarios, such as planning a savings goal where you deposit a fixed amount each week or predicting the growth of a stack of items in a warehouse. For example, if you start with 3 crates and add 4 crates each week, you can easily calculate how many crates you will have in a year (52 weeks) using the formula \(a_n = 3 + (n-1) \times 4\). Algebraic products are similarly useful in fields like engineering or physics when calculating the area of complex shapes or analyzing forces in a system where variables represent dimensions or intensities. Mastery of these patterns simplifies complex real-world problems into solvable mathematical expressions.

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