1. **State the problem:** We need to find the pattern rule in each series and complete the missing terms. 2. **Series (a):** 32.4, 32.8, 33.2, 33.6, 34, 34.4 - The difference between consecutive terms is $32.8 - 32.4 = 0.4$. - This is an arithmetic sequence with common difference $d = 0.4$. - The series is already complete with 6 terms. 3. **Series (b):** [blank], 2.008, 2.013, [blank], [blank], 2.028 - Given terms: $a_2 = 2.008$, $a_3 = 2.013$, $a_6 = 2.028$. - Find the common difference $d$ assuming arithmetic progression: $$d = a_3 - a_2 = 2.013 - 2.008 = 0.005$$ - Use $a_n = a_1 + (n-1)d$ to find $a_1$: $$2.008 = a_1 + (2-1)\times 0.005 \Rightarrow a_1 = 2.008 - 0.005 = 2.003$$ - Find missing terms: $$a_4 = a_1 + 3d = 2.003 + 3 \times 0.005 = 2.018$$ $$a_5 = a_1 + 4d = 2.003 + 4 \times 0.005 = 2.023$$ 4. **Series (c):** 7, 7.5, 6.5, [blank], [blank], 5.75 - Observe the pattern: - $7$ to $7.5$ is $+0.5$ - $7.5$ to $6.5$ is $-1.0$ - The pattern alternates adding $0.5$ then subtracting $1.0$. - Continue the pattern: $$a_4 = 6.5 + 0.5 = 7.0$$ $$a_5 = 7.0 - 1.0 = 6.0$$ 5. **Final completed series:** - (a) $32.4, 32.8, 33.2, 33.6, 34, 34.4$ - (b) $2.003, 2.008, 2.013, 2.018, 2.023, 2.028$ - (c) $7, 7.5, 6.5, 7.0, 6.0, 5.75$