4.4 Q-7 | Sequence And Series | Mathematics 11

Exercise 4.4 — Question 7

Note: The original problem statement for Q-7 of Exercise 4.4 was not provided. The solution framework below is based on the linked SLOs covering arithmetic-geometric sequences (M-11-A-27), harmonic sequences (M-11-A-25), and summation formulas (M-11-A-26). Please insert the exact question text from the FBISE textbook.

Key Formulas for This Exercise

Arithmetic-Geometric Sequence

An arithmetic-geometric sequence is formed by multiplying corresponding terms of an arithmetic sequence and a geometric sequence.

If the arithmetic sequence has first term $a$ and common difference $d$ , and the geometric sequence has first term $1$ and common ratio $r$ , then the general term is:

$T_{n} = [a + (n - 1) d] r^{n - 1}$

Sum to $n$ terms:

$S_{n} = \frac{a - [ a + ( n - 1 ) d ] r ^{n}}{1 - r} + \frac{d r ( 1 - r ^{n - 1} )}{( 1 - r ) ^{2}}, r \neq = 1$

Sum to infinity (when $∣ r ∣ < 1$ ):

$S_{\infty} = \frac{a}{1 - r} + \frac{d r}{( 1 - r ) ^{2}}$

Summation Formulas (Sigma Notation)

Series	Formula
$k = 1 \sum n k$	$\frac{n ( n + 1 )}{2}$
$k = 1 \sum n k^{2}$	$\frac{n ( n + 1 ) ( 2 n + 1 )}{6}$
$k = 1 \sum n k^{3}$	$[\frac{n ( n + 1 )}{2}]^{2}$

Harmonic Sequence

A sequence $a_{1}, a_{2}, a_{3}, \dots$ is a harmonic sequence if the reciprocals $\frac{1}{a _{1}}, \frac{1}{a _{2}}, \frac{1}{a _{3}}, \dots$ form an arithmetic sequence.

The $n$ -th term of a harmonic sequence: $a_{n} = \frac{1}{a + ( n - 1 ) d}$ where $a$ and $d$ are the first term and common difference of the corresponding arithmetic sequence.

Solution

(Insert the worked solution for Q-7 here once the problem statement is confirmed from the FBISE textbook.)

Exercise 4.4 — Question 7

Key Formulas for This Exercise

Arithmetic-Geometric Sequence

An arithmetic-geometric sequence is formed by multiplying corresponding terms of an arithmetic sequence and a geometric sequence.

If the arithmetic sequence has first term $a$ and common difference $d$ , and the geometric sequence has first term $1$ and common ratio $r$ , then the general term is:

$T_{n} = [a + (n - 1) d] r^{n - 1}$

Sum to $n$ terms:

$S_{n} = \frac{a - [ a + ( n - 1 ) d ] r ^{n}}{1 - r} + \frac{d r ( 1 - r ^{n - 1} )}{( 1 - r ) ^{2}}, r \neq = 1$

Sum to infinity (when $∣ r ∣ < 1$ ):

$S_{\infty} = \frac{a}{1 - r} + \frac{d r}{( 1 - r ) ^{2}}$

Summation Formulas (Sigma Notation)

Series	Formula
$k = 1 \sum n k$	$\frac{n ( n + 1 )}{2}$
$k = 1 \sum n k^{2}$	$\frac{n ( n + 1 ) ( 2 n + 1 )}{6}$
$k = 1 \sum n k^{3}$	$[\frac{n ( n + 1 )}{2}]^{2}$

Harmonic Sequence

A sequence $a_{1}, a_{2}, a_{3}, \dots$ is a harmonic sequence if the reciprocals $\frac{1}{a _{1}}, \frac{1}{a _{2}}, \frac{1}{a _{3}}, \dots$ form an arithmetic sequence.

The $n$ -th term of a harmonic sequence: $a_{n} = \frac{1}{a + ( n - 1 ) d}$ where $a$ and $d$ are the first term and common difference of the corresponding arithmetic sequence.

Solution

(Insert the worked solution for Q-7 here once the problem statement is confirmed from the FBISE textbook.)