Question

how many series 1,4,7,10 amount in 176

Answer 1

Given:

The series 1,4,7,10.

Sum of $n$ terms, $S_n=176$

To find:

The value of $n$ for having the given sum.

Solution:

First of all, let us have a look at the series.

First term, $a$ = 1

Common difference, d = Second term - First term = 4 - 1 = 3

Each term has a difference equal to 3.

Therefore, the given series is an Arithmetic Progression (AP).

Formula for sum of $n$ terms is given as:

$S_n = \dfrac{n}{2}(2a+(n-1)d)$

Putting the given values:

$176 = \dfrac{n}{2} (2\times 1+(n-1)3)\\\Rightarrow 352 = n(2+3n-3)\\\Rightarrow 352 = n(3n-1)\\\Rightarrow 3n^2-n-352=0\\\Rightarrow 3n^2-33n+32n-352=0\\\Rightarrow 3n(n-11)+32(n-11)=0\\\Rightarrow (3n+32)(n-11)=0\\\Rightarrow n=11, -\frac{32}{3}$

Value of $n$ can not be negative.

Therefore, the answer is 11.