1) Find the explicit formula: 5, 14, 23, 32, 41, ...
a1 = 5, d = 14 − 5 = 9, an = a1 + d (n – 1)
⇒ an = 5 + 9 (n – 1)
2) Find the explicit formula: −7, −2, 3, 8, 13, ...
a1 = −7, d = −2 − (−7) = 5, an = a1 + d (n – 1)
⇒ an = −7 + 5 (n – 1)
3) Find the explicit formula and a12: −22, −16, −10, −4, 2, ...
a1 = −22, d = −16 − (−22) = 6, an = a1 + d (n – 1)
⇒ an = −22 + 6 (n – 1) ⇒ a12 = −22 + 6 (12 – 1) = 44
4) Find the explicit formula and a21: a30 = 72, d = −5
a30 = 72, d = −5, an = a1 + d (n – 1)
⇒ a30 = a1 + (−5) (30 – 1) ⇒ a1 = a30 + 5(29) = 72 + 145 = 217
⇒ an = 217 − 5 (n – 1) ⇒ a21 = 217 − 5(20) = 117
5) Find the first 10 terms: a15 = 50, d = 4.5
a15 = 50, d = 4.5, an = a1 + d (n – 1)
⇒ a15 = a1 + 4.5 (15 – 1) ⇒ a1 = 50 − 4.5(14) = −13
⇒−13, −8.5, −4, 0.5, 5, 9.5, 14, 18.5, 23, 27.5, ...
6) Find the first 7 terms: a41 = 178, d = 4
a41 = 178, d = 4, an = a1 + d (n – 1)
⇒ a41 = a1 + 4 (41 – 1) ⇒ a1 = 178 − 4(40) = 18
⇒18, 22, 26, 30, 34, 38, 42, ...
7) Find a30: 45, , 41.8, 38.6, 35.4, ...
a1 = 45, d = 41.8 − 45 = −3.2, an = a1 + d (n – 1)
⇒ an = 45 − 3.2 (n – 1) ⇒ a30 = 45 − 3.2 (30 – 1) = −47.8
8) Find a27: a41 = 167, d = 6.4
a41 = 167, d = 6.4, an = a1 + d (n – 1)
⇒ a41 = a1 + 6.4 (41 – 1) ⇒ a1 = 167 − 6.4(40) = −98
⇒ an = −98 + 6.4 (n – 1) ⇒ a27 = −98 + 6.4(26) = 68.4
9) Find the explicit formula and the sum of the first five terms: a1 = 10, d = 3
a1 = 10, d = 3, an = a1 + d (n – 1)
⇒ an = 10 + 3 (n – 1)
∑n − 1k = 0 (a + kd) = n2 (2a + (n − 1)d)
⇒ ∑4k = 0 (10 + 3k) = 52 (20 + (4)3) = 52×(32) = 80
10) Find the explicit formula and the sum of the first 16 terms: −55, −48, −41, −34, −27, ...
a1 = −55, d = −48 − (−55) = 7, an = a1 + d (n – 1)
⇒ an = −55 + 7 (n – 1)
∑n − 1k = 0 (a + kd) = n2 (2a + (n − 1)d)
⇒ ∑15k = 0 (−55 + 7k) = 162 (−110 + (15)7) = 8×(−5) = −40