The 3^{\text{rd}} and 6^{\text{th}} terms of an Arithmetic Progression (A.P.) are 13 and 2… — JAMB UTME

The $3^{\text{rd}}$ and $6^{\text{th}}$ terms of an Arithmetic Progression (A.P.) are $13$ and $28$ respectively. What is the sum of the first $10$ terms of the progression?

$255$ Answer
B
$280$
C
$205$
D
$48$

Answer

The sum of the first

10

terms of the progression is

255

The

3^{\text{rd}}

term is

a + 2d = 13

and the

6^{\text{th}}

term is

a + 5d = 28

. Subtracting these equations gives

3d = 15

, so

d = 5

, which leads to

a = 3

. Using the sum formula

S_{10} = \frac{10}{2}[2(3) + (10 - 1)5]

, we obtain

5(6 + 45) = 255

Step-by-Step Solution

Set up equations for the given terms using the nth term formula

T_n = a + (n - 1)d

a + 2d = 13

and

a + 5d = 28

The

3^{\text{rd}}

term corresponds to

n=3

and the

6^{\text{th}}

term corresponds to

n=6

Solve the simultaneous equations for

a

(first term) and

d

(common difference).

Subtracting the first equation from the second gives

3d = 15 \Rightarrow d = 5

. Substituting

d = 5

into the first equation yields

a + 2(5) = 13 \Rightarrow a = 3

To find any property of an A.P., the first term

a

and common difference

d

must be determined.

Calculate the sum of the first

10

terms using

S_n = \frac{n}{2}[2a + (n - 1)d]

S_{10} = \frac{10}{2}[2(3) + (10 - 1)5] = 5[6 + 45] = 5(51) = 255

Applying the formula for the sum of the first

n

terms with

n = 10

a = 3

, and

d = 5

Key Concept

Finding the sum of the first

n

terms of an Arithmetic Progression given two specific terms.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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