CALCU-The thirteenth term of an arithmetic progres

The thirteenth term of an arithmetic progression is 12 and the sum of the first 30 terms is -15 . Find the sum of the first 50 terms of the progression. [5]

Solution by Steps

step 1

Let's denote the first term of the arithmetic progression as

a_1

and the common difference as

d

. The

n

th term of an arithmetic progression is given by

a_n = a_1 + (n-1)d

step 2

Given that the 13th term

a_{13}

is 12, we can write the equation

a_{13} = a_1 + 12d = 12

step 3

The sum of the first

n

terms of an arithmetic progression is given by

S_n = \frac{n}{2}(2a_1 + (n-1)d)

step 4

Given that the sum of the first 30 terms

S_{30}

is -15, we can write the equation

S_{30} = \frac{30}{2}(2a_1 + 29d) = -15

step 5

We now have a system of two equations:

a_1 + 12d = 12

and

15(2a_1 + 29d) = -15

. We can solve this system to find

a_1

and

d

step 6

Solving the system, we find the values of

a_1

and

d

step 7

Once we have

a_1

and

d

, we can find the sum of the first 50 terms

S_{50}

using the formula from step 3 with

n = 50

step 8

Calculate

S_{50} = \frac{50}{2}(2a_1 + 49d)

using the found values of

a_1

and

d

Answer

[Insert final answer here]

Key Concept

Arithmetic Progression Formulas

Explanation

To find the sum of the first 50 terms, we use the formulas for the

n

th term and the sum of the first

n

terms of an arithmetic progression. We first determine the first term and the common difference by solving a system of equations derived from the given information.

What is the formula to find the sum of the first n terms of an arithmetic progression?

AskSia

Plus