Show that when a string fixed at its two ends vibrates in 1 loop, 2 loops, 3 loops and 4 loops, the frequencies are in the ratio 1:2:3:4.

Hint: Use the derivation of standing waves.
Step 1: Find the frequency of n-harmonic.
Let, there are n numbers of loops in the string.
Length corresponding each loop is λ2.
Now, we can write,   
                                        L=2λ=2Ln                     for n loops
                                     ν=n2Lv=n2LTμ   velocity of transverse wave =T/μ
 νn                                                                      length and speed are constants
Step 2: Find the ratio of frequencies.
So,                  ν1:ν2:ν3:ν4=n1:n2:n3:n4