Key Concepts and Formulas

• Arithmetic Progression (A.P.): A sequence where the difference between consecutive terms is constant. This constant is called the common difference ($d$).
• $n^{th}$ term of an A.P.: $a_n = a + (n-1)d$, where $a$ is the first term.
• Sum of the first $n$ terms of an A.P.: $S_n = \frac{n}{2}[2a + (n-1)d]$.

Step-by-Step Solution

Step 1: Formulate equations using the given information

We are given $S_n = 700$, $a_6 = 7$, and $S_7 = 7$. We will use these to form equations involving the first term ($a$) and the common difference ($d$).

1. Using $S_7 = 7$: We apply the formula for the sum of the first $n$ terms: $S_7 = \frac{7}{2}[2a + (7-1)d]$ Substitute $S_7 = 7$: $7 = \frac{7}{2}[2a + 6d]$ Multiply both sides by $\frac{2}{7}$: $7 \times \frac{2}{7} = 2a + 6d$ $2 = 2a + 6d$ Divide by 2: $1 = a + 3d \quad \text{(Equation 1)}$ Reasoning: This equation establishes a relationship between the first term and the common difference based on the sum of the first 7 terms.

2. Using $a_6 = 7$: We apply the formula for the $n^{th}$ term: $a_6 = a + (6-1)d$ Substitute $a_6 = 7$: $7 = a + 5d \quad \text{(Equation 2)}$ Reasoning: This equation establishes another relationship between the first term and the common difference based on the 6th term.

Step 2: Solve the system of linear equations for $a$ and $d$

We now have a system of two linear equations:

1. $a + 3d = 1$
2. $a + 5d = 7$

Subtract Equation 1 from Equation 2: $(a + 5d) - (a + 3d) = 7 - 1$ $2d = 6$ Divide by 2: $d = 3$ Reasoning: By subtracting the two equations, we eliminate 'a' and can directly solve for the common difference 'd'.

Now, substitute the value of $d$ into Equation 1 to find $a$: $a + 3(3) = 1$ $a + 9 = 1$ $a = 1 - 9$ $a = -8$ Reasoning: Substituting the found value of 'd' back into one of the original equations allows us to solve for the first term 'a'.

Step 3: Determine the value of $n$ using the sum $S_n = 700$

We have found $a = -8$ and $d = 3$. Now we use the given $S_n = 700$ to find the value of $n$. Apply the sum formula: $S_n = \frac{n}{2}[2a + (n-1)d]$ Substitute $S_n = 700$, $a = -8$, and $d = 3$: $700 = \frac{n}{2}[2(-8) + (n-1)3]$ $700 = \frac{n}{2}[-16 + 3n - 3]$ $700 = \frac{n}{2}[3n - 19]$ Multiply both sides by 2: $1400 = n(3n - 19)$ $1400 = 3n^2 - 19n$ Rearrange into a quadratic equation: $3n^2 - 19n - 1400 = 0$ Reasoning: We use the sum formula with the known values of $a$ and $d$ to set up a quadratic equation in terms of $n$. Solving this quadratic equation will give us the number of terms.

We can solve this quadratic equation by factoring or using the quadratic formula. Let's try factoring. We look for two numbers that multiply to $3 \times -1400 = -4200$ and add to $-19$. These numbers are $-75$ and $56$. $3n^2 - 75n + 56n - 1400 = 0$ Factor by grouping: $3n(n - 25) + 56(n - 25) = 0$ $(3n + 56)(n - 25) = 0$ This gives two possible values for $n$: $3n + 56 = 0 \implies n = -\frac{56}{3}$ (Not a valid term number as $n$ must be a positive integer) $n - 25 = 0 \implies n = 25$ Reasoning: Solving the quadratic equation yields possible values for $n$. Since $n$ represents the number of terms, it must be a positive integer. Thus, $n=25$ is the only valid solution.

Step 4: Calculate $a_n$ using the found values of $n$, $a$, and $d$

We need to find $a_n$, and we have found $n=25$, $a=-8$, and $d=3$. Using the $n^{th}$ term formula: $a_n = a + (n-1)d$ Substitute the values: $a_{25} = -8 + (25-1)3$ $a_{25} = -8 + (24)3$ $a_{25} = -8 + 72$ $a_{25} = 64$ Reasoning: With all the necessary parameters ($a$, $d$, and $n$) determined, we can now calculate the value of the $n^{th}$ term.

Common Mistakes & Tips

• Sign errors: Be very careful with negative signs when substituting values for $a$ and $d$, and during algebraic manipulations.
• Formula recall: Ensure you have the correct formulas for $a_n$ and $S_n$ memorized.
• Solving quadratic equations: Double-check your factoring or use of the quadratic formula to avoid errors in finding $n$. Remember that $n$ must be a positive integer.

Summary

The problem provided information about the sum of terms ($S_n$ and $S_7$) and a specific term ($a_6$) of an Arithmetic Progression. By using the standard formulas for the $n^{th}$ term and the sum of an A.P., we set up a system of linear equations to find the first term ($a$) and the common difference ($d$). Subsequently, we used the given sum $S_n = 700$ to form and solve a quadratic equation for $n$. Finally, with $a$, $d$, and $n$ determined, we calculated the value of $a_n$.

The final answer is \boxed{64}.