If an arithmetic progression has (a=20), (d=-3), and (n=7), what is the sum of the first (7) terms?

(77)

For an arithmetic progression, the first term is \\(a=20\\), the common difference is \\(d=-3\\), and the number of terms is \\(n=7\\). The seventh term is \\(a_7=a+(7-1)d=20+6(-3)=2\\). The sum formula is \\(S_n=\frac{n}{2}[2a+(n-1)d]\\), so \\(S_7=\frac{7}{2}[40+6(-3)]=\frac{7}{2}(22)=77\\).

The same result comes from pairing the first and last terms: the average is \\(\frac{20+2}{2}=11\\), and seven terms give \\(7\times11=77\\). Thus option A is correct. The negative difference decreases each term, but it must be included with its sign; replacing \\(d=-3\\) by 3 would produce an incorrect sum.

This is a Class 10 Mathematics question. Chapter: Arithmetic Progressions (AP). Topic: Finding the sum of the first (n) terms of an AP.