Arithmetic and geometric progressions

Learning objectives

Recognise an arithmetic progression and find the first term and common difference, then use uₙ = a + (n − 1)d and Sₙ = ½n(2a + (n − 1)d) to find a term, a sum or the number of terms

Recognise a geometric progression and find the first term and common ratio, then use uₙ = ar^(n − 1) and Sₙ = a(1 − rⁿ)/(1 − r) to find a term, a sum or the number of terms

State the condition that the modulus of r is less than 1 for a geometric series to converge, and use S∞ = a/(1 − r) to find a sum to infinity, explaining why no sum to infinity exists when |r| ≥ 1

Form and solve simultaneous equations from information about two terms or two sums of a progression, including cases where a logarithm is needed to find n

Apply progressions to practical contexts such as repayments, savings with a fixed percentage increase or the total distance travelled by a bouncing ball, and interpret the answer in context