1. The last term is 5.
2. First term: 3/2, common ratio: 2/3 or first term: 2/3, common ratio: 3/2; terms: 3/2, 1, 2/3 or 2/3, 1, 3/2.
3. Proof: Since a, b, c are in G.P., b² = ac. Then (a² + b²)(b² + c²) = (ab + bc)², so a² + b², ab + bc, b² + c² are in G.P.
4. Proof: Since a², b², c² are in A.P., 2b² = a² + c². Then 2/(b+c) = 1/(c+a) + 1/(a+b), so 1/(b+c), 1/(c+a), 1/(a+b) are in A.P.
5. The 50th number is 1/50.
6. (i) 4th term (5 - 0i) is purely real. (ii) 3rd term (6 - 2i) is purely imaginary.
7. Sum of first mn terms is (mn + 1)/2.
8. Total cost is Rs. 26,400.
9. Proof: Let first term be a and common difference d. Sum of m terms / Sum of n terms = m²/n² implies (2a + (m-1)d)/(2a + (n-1)d) = m/n. Solving gives the ratio of mth to nth term as (2m-1):(2n-1).
Parent Tip: Review the logic above to help your child master the concept of series worksheet.
