Understanding the Problem of an Arithmetic Progression

Given an arithmetic progression (AP) where the first term a1 is 2, the nth term an is 70, and the sum of the first n terms, (S_n), is 3741, this article will walk you through the steps to find the value of n.

Step-by-Step Solution

First, let's establish the necessary equations for an arithmetic progression (AP). An AP is defined by the first term a and the common difference d. For the nth term, the formula is:

[ a_n a (n-1)d ]

Additionally, the sum of the first n terms in an AP is given by:

[ S_n frac{n}{2} (2a (n-1)d) ]

Given:

a1 2

(a_n 70)

Sn 3741

Step 1: Use the formula for the nth term to find the common difference d.

[ 70 2 (n-1)d ]

Step 2: Solve the equation for d.

[ (n-1)d 68 ]

Step 3: Use the sum formula to find n.

[ S_n frac{n}{2} (2a (n-1)d) ]

Substitute the known values:

[ 3741 frac{n}{2} (4 68) ]

Simplify and solve for n:

[ 3741 36n ]

[ n frac{3741}{36} 103.917 approx 104 ]

Since n must be a whole number, the correct value of n is 104.

Mathematical Verification

Substitute n 104 back into the equation for the common difference d to confirm:

[ 103d 68 ]

[ d frac{68}{103} approx 0.66 ]

This value of d is consistent, confirming that the value of n is indeed 104.

The Arithmetic Sequence

Given the first term a 2 and the common difference d 68/103, the arithmetic sequence can be written as:

[ t_n 2 (n-1) frac{68}{103} ]

This formula can be simplified to:

[ t_n frac{68n 138}{103} ]

A specific term in the sequence can be calculated by substituting the value of n. For example, the 104th term is:

[ t_{104} 2 103 frac{68}{103} 70 ]

Thus, the value of n is 104.

Conclusion

The value of n in an arithmetic progression where the first term is 2, the nth term is 70, and the sum of the first n terms is 3741 is 104. The detailed steps above demonstrate the mathematical approach to solving such problems.