Question

The algebraic expression of an arithmetic sequence is 7n+3. What is its 100th term?

Answer 1

Answer :

We have the algebraic expression of an arithmetic sequence, i.e. 7n + 3. By substituting 1, 2 and 3 in place of n in the expression. We will get the value of the first, second and third term of the arithmetic progression. And then by using the formula of arithmetic progression, we will get our required answer.

Solution :

Given :

• Algebraic expression of an arithmetic progression = 7n + 3

To find :

• The 100th term of arithmetic progression

Solution :

$\\ \longrightarrow \sf t_1 = 7n + 3$

$\\ \longrightarrow \sf t_1 = 7(1) + 3$

$\\ \longrightarrow \sf t_1 = 7 + 3$

$\\ \longrightarrow \sf t_1 = 10$

$\\ \longrightarrow \sf t_2 = 7n + 3$

$\\ \longrightarrow \sf t_2 = 7(2) + 3$

$\\ \longrightarrow \sf t_2 = 14 + 3$

$\\ \longrightarrow \sf t_2 = 17$

$\\ \longrightarrow \sf t_3 = 7n + 3$

$\\ \longrightarrow \sf t_3 = 7(3) + 3$

$\\ \longrightarrow \sf t_3 = 21 + 3$

$\\ \longrightarrow \sf t_3 = 24$

The first term of A.P (a) = 10

Calculating the common difference :

Formula to calculate the common difference :

$\star \quad \underline{ \boxed{ \ \boldsymbol{ \red{d = t_2 - t_1}}}}$

where,

• d denotes the common difference
• $\textsf{ t_2 \: denotes the second term}$
• $\textsf{ t_1 \: denotes the first term}$

Substituting the given values :

$\\ \longrightarrow \sf d = 17 - 10$

$\\ \longrightarrow \sf d = 7$

Therefore, the common difference (d) = 7

Calculating the 100th term of A.P :

Formula to calculate the terms of A.P :

$\underline{\boxed{\boldsymbol{\red{t_n = a + (n - 1)d}}}}$

Substituting the given values :

$\\ \longrightarrow \sf t_{100} = 10 + (100 - 1)7$

$\\ \longrightarrow \sf t_{100} = 10 + (99)7$

$\\ \longrightarrow \sf t_{100} = 10 + 99 \times 7$

$\\ \longrightarrow \sf t_{100} = 10 + 693$

$\\ \longrightarrow \sf t_{100} = 703$

$\underline{\textsf{\textbf{The 100th term of A.P. = 703}}}$