solve the question:---**Question Stem:**
Let $a_n$ be the $n^{th}$ term of an A.P. If $S_n = a_1 + a_2 + a_3 + \dots + a_n = 700$, $a_6 = 7$ and $S_7 = 7$, then $a_n$ is equal to :
**Options:**
A 65
B 56
C 70
D 64
**Other Relevant Text:**
None.
**Chart/Diagram Description:**
None.
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We have an arithmetic progression problem. Given that the sum of n terms equals 700, the 6th term equals 7, and the sum of first 7 terms equals 7. We need to find the value of the nth term. Let's start by setting up the equations using the arithmetic progression formulas.
Now let's set up our equations. For an arithmetic progression, the nth term is a₁ plus n minus 1 times d. So a₆ equals a₁ plus 5d equals 7. For the sum formula, S₇ equals 7 over 2 times 2a₁ plus 6d, which equals 7. Simplifying this gives us a₁ plus 3d equals 1. Now we have a system of two equations with two unknowns.
Let's solve this system of equations. We subtract the second equation from the first: a₁ plus 5d minus a₁ plus 3d equals 7 minus 1. This simplifies to 2d equals 6, so d equals 3. Now we substitute d equals 3 back into the second equation: a₁ plus 3 times 3 equals 1, which gives us a₁ plus 9 equals 1, so a₁ equals negative 8. We now have our first term and common difference.
Now we use the condition that Sₙ equals 700 to find n. Using the sum formula with a₁ equals negative 8 and d equals 3, we get 700 equals n over 2 times 2 times negative 8 plus n minus 1 times 3. Simplifying, we get 700 equals n over 2 times 3n minus 19. Multiplying both sides by 2 gives us 1400 equals n times 3n minus 19, which expands to the quadratic equation 3n² minus 19n minus 1400 equals 0. Using the quadratic formula, we get n equals 25 or n equals negative 112 over 6. Since n must be positive, n equals 25.
Finally, let's calculate the nth term for n equals 25. Using the formula aₙ equals a₁ plus n minus 1 times d, we substitute our values: a₂₅ equals negative 8 plus 25 minus 1 times 3, which equals negative 8 plus 24 times 3, which equals negative 8 plus 72, giving us 64. We can verify this is correct by checking that S₂₅ equals 700. Therefore, aₙ equals 64, which corresponds to answer choice D.