Q.Find the position of the image formed by the lens combination given in the Fig. 9.20.

Apply the thin-lens formula $\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$ lens by lens, using each image as the object for the next. Lens 1 ($f=+10$, $u=-30$) gives $v_1=+15$ cm; measured from lens 2 (5 cm away) the object is $u_2=+10$ cm, and with $f_2=-10$ this gives $v_2=\infty$ (parallel beam). Parallel rays into lens 3 …

Concept understanding

For thin lenses in a row on one axis, refraction is handled one lens at a time with

using the image of one lens as the object for the next and accounting for the separation between lenses. Sign convention: distances measured along the incident light (left$\to$right) are positive; convex $f>0$, concave $f<0$.

Step-by-step (setup: object 30 cm left of lens 1; lenses at 0, 5 cm, 10 cm with $f_1=+10$, $f_2=-10$, $f_3=+30$)

Lens 1 (convex, $f_1=+10$ cm): $u_1=-30$ cm.

A real image forms 15 cm to the right of lens 1.

Lens 2 (concave, $f_2=-10$ cm): lens 2 is 5 cm right of lens 1, so this image is $15-5=10$ cm to the right of lens 2 — a virtual object, $u_2=+10$ cm.

$\frac{1}{v_2}=\frac{1}{f_2}+\frac{1}{u_2}=-\frac{1}{10}+\frac{1}{10}=0\;\Rightarrow\; v_2=\infty.$ …

Method: Tracing an Image Through a System of Thin Lenses

This method solves any problem with two or more optical elements (lenses, or a lens combined with a mirror) placed one after another along a common axis.

Steps

Step 1: List every element's focal length and its separation from the next

Write down $f_1, f_2, f_3, \dots$ with correct signs (convex $f>0$, concave $f<0$) and the physical distance between consecutive elements. Fix a sign convention once (incident-light direction positive) and use it throughout.

Step 2: Apply the thin-lens equation to the first element using the real object distance

Solve for $v_1$, the image distance measured from lens 1.

Step 3: Convert that image into the object for the next element

The image formed by one element becomes the object for the next — but its object distance must be measured from the next element, not from the first. Subtract the inter-lens separation from $v_1$ to get the raw position, then decide the sign of $u_2$: if the still-unformed image lies beyond the next element (light hasn't converged yet), it is a virtual object ($u_2>0$ under this convention); if it lies before, it's a normal real object. …