Exam Practice Worksheet

Progressions • SPM Paper 1 & 2

SPM KSSM Diagnostic Drill

Form 4 Chapter 5: Progressions (Janjang)

Total Marks: 40 Marks • Time Allowed: 50 Minutes

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A Bahagian A: SPM Paper 1 Format [16 Marks]

Answer all questions
Soalan 1 [4 Marks]

The sum of the first $n$ terms of an arithmetic progression is given by $S_n = 3n^2 + 5n$.

(a) Find the first term, $a$, and the common difference, $d$. [3 marks]

(b) Find the 10th term, $T_{10}$. [1 mark]

Soalan 2 [4 Marks]

The first term of a geometric progression is $500$ and its common ratio is $0.8$. Find the smallest value of $n$ such that the sum of the first $n$ terms exceeds $2,400$.

Soalan 3 (Paper 1 Sec B) [8 Marks]

A geometric progression has first term $a$ and common ratio $r$. It is given that the sum to infinity is $16$ and the second term is $3$.

(a) Show that $16r^2 - 16r + 3 = 0$. [3 marks]

(b) Hence, find the two possible sets of values for $a$ and $r$. [3 marks]

(c) If all terms in the progression are positive and strictly decreasing, calculate the sum of the first 6 terms. [2 marks]

B Bahagian B: SPM Paper 2 Format & Real-World KBAT [24 Marks]

Answer all questions
Soalan 4 (Paper 2 Standard) [10 Marks]

In an arithmetic progression, the 4th term is $19$ and the sum of the first 8 terms is $172$.

(a) Find the first term, $a$, and the common difference, $d$. [4 marks]

(b) Calculate the sum of terms from the 9th term to the 25th term inclusive. [3 marks]

(c) The 1st term, 4th term, and 16th term of this arithmetic progression form three consecutive terms of a geometric progression. Verify whether this statement is mathematically valid. [3 marks]

Soalan 5 (KBAT Financial Retirement Accumulation Modeling) [8 Marks]
Encik Danial plans for his daughter's university education fund by depositing money into a special trust fund on the first day of every year for $18$ consecutive years. He starts with an initial deposit of RM 2,000 on Year 1. He considers two financial contribution schemes:

Plan A (Arithmetic Progression): The annual deposit increases by RM 250 each subsequent year.

Plan B (Geometric Progression): The annual deposit increases by $8\%$ each subsequent year.

(a) Calculate the deposit made by Encik Danial in the 18th year under Plan A. [2 marks]

(b) Calculate the deposit made by Encik Danial in the 18th year under Plan B. [2 marks]

(c) Calculate the total savings deposited over the entire 18 years for both Plan A and Plan B. [3 marks]

(d) Based on your calculations in (c), recommend which plan Encik Danial should choose to maximize the fund. [1 mark]

Soalan 6 (Non-Routine Synthesis) [6 Marks]

A sequence of squares $S_1, S_2, S_3, \dots$ is constructed such that the vertices of square $S_{k+1}$ are the midpoints of the sides of square $S_k$. The side length of the first square $S_1$ is $16\text{ cm}$.

(a) Show that the areas of the squares form a geometric progression and state the common ratio. [3 marks]

(b) Calculate the sum to infinity of the areas of all the squares. [3 marks]