Math Problem Statement

Charlie deposited $115 in a savings account. Each week thereafter, he deposits $35 into the account. (a) Write a formula to represent this sequence. (b) How much total money has Charlie deposited after 30 weeks?

Solution

Let's solve the problem step by step.

Part (a): Writing a formula for the sequence

Charlie initially deposits $115 into the account. Each week thereafter, he deposits $35. This forms an arithmetic sequence where:

  • The first term aa is 115115.
  • The common difference dd is 3535.

The general formula for an arithmetic sequence is:

an=a+(n1)da_n = a + (n - 1) \cdot d

Where:

  • ana_n is the amount deposited after nn weeks,
  • aa is the initial amount ($115),
  • dd is the amount added each week ($35).

So, the formula becomes:

an=115+(n1)35a_n = 115 + (n - 1) \cdot 35

Part (b): Total money deposited after 30 weeks

To find the total deposited after 30 weeks, we need to sum the sequence up to n=30n = 30.

The sum SnS_n of the first nn terms of an arithmetic sequence is:

Sn=n2(2a+(n1)d)S_n = \frac{n}{2} \cdot (2a + (n - 1) \cdot d)

Plugging in the values:

  • n=30n = 30
  • a=115a = 115
  • d=35d = 35

S30=302(2115+(301)35)S_{30} = \frac{30}{2} \cdot (2 \cdot 115 + (30 - 1) \cdot 35) S30=15(230+1015)S_{30} = 15 \cdot (230 + 1015) S30=151245S_{30} = 15 \cdot 1245 S30=18675S_{30} = 18675

So, the total money deposited after 30 weeks is $18,675.

Would you like further details or have any questions?

Related Questions:

  1. What is the total amount deposited after 10 weeks?
  2. What if the initial deposit was $200 instead of $115? How would the formula change?
  3. How much would be deposited after 52 weeks (1 year)?
  4. How would the sum formula look if the deposit amount per week increased by $5 each month?
  5. Can you write the recursive formula for this sequence?

Tip:

Always identify whether a problem involves arithmetic or geometric sequences by examining whether the change between terms is constant (arithmetic) or multiplicative (geometric).

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Math Problem Analysis

Mathematical Concepts

Arithmetic Sequences
Algebra

Formulas

General formula for an arithmetic sequence: a_n = a + (n - 1) · d
Sum of first n terms of an arithmetic sequence: S_n = (n/2) · (2a + (n - 1) · d)

Theorems

Arithmetic Sequence Formula
Sum of Arithmetic Sequence Formula

Suitable Grade Level

Grades 7-9