Monkey Raptor

First, the denotational dilly

• Sum of n-finite series is usually denoted by $ S_n $.
• The $ n^{th} $ member (or term) is denoted by $ a_n $.
• The 1^st term is denoted by $ a $ (or to be consistent, $ a_1 $).
  Because I'm very not detergent, let's use just $ a $ instead.
• The difference between consecutive terms is denoted by $ d $ (as in difference).

Second, the finite Arithmetic formulas

These are the um, thing we probably forgot or not. As a reminder, they look like this:

• Arithmetic series (sum):

  $ S_n = n/2(2a + (n-1)d) $

• The progression member/term:

  $ a_n = a + (n-1)d $

If you don't believe me, try either one by substi...

Third, the concept

We need to expand the known variables using the formulas above, so then we'll generate a system of linear equations in two variables.

Let's then translate that

• $ S_10 = 110 $
• $ a_11 + a_13 = -4 $
• The question: $ a_1 + a_2 = ? $

➤ we need to find the $ a $ and $ d $, so then:

Expand the $ S_10 = 110 $

$ S_10 = 110 $

$ 10/2(2a + (10 - 1)d) = 110 $

$ 5(2a + 9d) = 110 $

$ 2a + 9d = 110/5 $

$ 2a + 9d = 22 ... (1) $ Our first linear equation

Expand the $ a_11 + a_13 = -4 $

$ a_11 + a_13 = -4 $

$ (a + (11 - 1)d) + (a + (13 - 1)d) = -4 $

$ (a + 10d) + (a + 12d) = -4 $

$ 2a + 22d = -4 ...(2)$ Second linear equation

Now we have 2 equations in 2 variables; we can find the value of each thing using basic elimination

$ 2a + 9d = 22 ... (1) $

$ 2a + 22d = -4 ... (2)$

------------------------ ^—

$ -13d = 26 $

$ d = -2 $

And then the $ a $; using substitution; let's take the 1^st equation

$ 2a + 9d = 22...(1)$

$ 2a + 9(-2) = 22 $

$ 2a + -18 = 22 $

$ 2a = 22 + 18 $

$ a = 40/2 $

$ a = 20 $ COMPLETE!

The question is $ a_1 + a_2 = ?$

Expand that into:

$ ⇔ (a + (1 - 1)d) + (a + (2 - 1)d)$

$ ⇔ (a) + (a + d)$

$ ⇔ 2a + d $

Substitute $ a = 20 $ and $ d = -2 $

$ ⇔ 2(20) + (-2) $

$ ⇔ 38 $

And the answer is 38.

WOOO!