Progression:

A progression is simply a sequence of numbers following a pattern. The nth term is the formula that helps you find any number in that sequence without writing out the whole list. Think of it like knowing the rule for arranging students in a queue.

In arithmetic progressions, numbers increase by a fixed amount. For example, if school fees increase by ₦5,000 yearly starting from ₦50,000, the nth term formula is: a_n = 50,000 + (n-1)×5,000. This tells you exactly what year five's fees will be without calculating each year separately.

In geometric progressions, numbers multiply by a constant. A bacteria population doubling daily follows this pattern. The nth term formula helps you predict the population on any day.

When you have a progression, you often need to find the total of all its terms—that's what "sum" means. For Arithmetic Progression (A.P.), where terms increase by a constant difference, use the formula S = n/2(first term + last term) or S = n/2[2a + (n-1)d]. Think of it like calculating total savings: if you save ₦500 in week 1, ₦1,000 in week 2, and ₦1,500 in week 3, the sum shows your total savings after three weeks.

For Geometric Progression (G.P.), where each term multiplies by a constant ratio, the formula is S = a(r^n - 1)/(r - 1). For example, if bacteria double daily starting with 10 cells, finding the sum helps you count total bacteria over several days.

When you have a geometric progression (G.P.) with a common ratio between -1 and 1, something magical happens: the terms get smaller and smaller until they almost disappear. The sum to infinity is the total you get when you add infinitely many terms together.

Think of it like this: imagine you're sharing ₦1,000 with friends. First person gets half (₦500), second gets half of that (₦250), third gets ₦125, and so on forever. Even though you keep sharing forever, you'll never give out more than ₦1,000 total! This is sum to infinity.

The formula is S∞ = a/(1 - r), where 'a' is the first term and 'r' is the common ratio. Remember this only works when |r| < 1.

The closure property means that when you perform an operation on numbers within a set, the result stays within that same set. Think of it like a family – if both parents are Nigerian, their children are also Nigerian. They don't suddenly become something else.

In arithmetic progressions, if you add or multiply terms from the sequence, the operation is "closed" within certain number sets. For example, in the sequence 2, 5, 8, 11, 14... all terms are natural numbers. When you add any two terms like 5 + 11 = 16, you still get a natural number.

Consider Nigerian trader Mrs. Adekunle selling oranges in arithmetic sequence: Day 1 she sells 10, Day 2 she sells 15, Day 3 she sells 20. If she adds any two days' sales, she gets whole oranges – the operation closes within whole numbers.