Finding the Sum of the First 21 Terms in an Arithmetic Progression

Arithmetic Progressions (AP) are a fundamental concept in mathematics, where each term after the first is obtained by adding a constant, known as the common difference (d). This article delves into how to find the sum of the first 21 terms of an AP given specific conditions, specifically when the 5th term is 16 and the 12th term is 37.

Step-by-Step Solution

To solve the problem, we start by setting up the equations for the 5th and 12th terms of the AP. For an arithmetic sequence, the nth term ( T_n ) is given by:

[ T_n a (n - 1)d ]

Given that the 5th term ( T_5 16 ) and the 12th term ( T_{12} 37 ), we can write:

[ a 4d 16 quad text{(1)} ] [ a 11d 37 quad text{(2)} ]

By subtracting equation (1) from equation (2), we can find the common difference:

[ (a 11d) - (a 4d) 37 - 16 ] [ 7d 21 ] [ d 3 ]

With the common difference known, we substitute ( d 3 ) into equation (1) to find the first term ( a ):

[ a 4 cdot 3 16 ] [ a 12 16 ] [ a 4 ]

Now that we have both the first term ( a 4 ) and the common difference ( d 3 ), we can find the 21st term ( T_{21} ):

[ T_{21} a 20d 4 20 cdot 3 64 ]

The sum of the first 21 terms ( S_{21} ) of an AP can be calculated using the formula:

[ S_n frac{n}{2} cdot (2a (n - 1)d) quad text{or} quad S_n frac{n}{2} cdot (a T_n) ]

Substituting ( n 21 ), ( a 4 ), and ( T_{21} 64 ) into the formula:

[ S_{21} frac{21}{2} cdot (4 64) frac{21}{2} cdot 68 21 cdot 34 714 ]

Conclusion

The sum of the first 21 terms in the given arithmetic progression is 714. Understanding the formula for common differences and the sum of terms is crucial in solving such problems. By applying these principles, we can easily find the desired sum of the first 21 terms.

Further Reading

For a deeper understanding of arithmetic progressions and other related concepts, you may want to explore:

Arithmetic Progression Formulas Common Differences in AP Sum of Terms in AP